Is it worth buying a lighter bike? If so what kind?

I would recommend you wait for one reason Covid. In most parts of the world bikes are in limited supply. Prices even on used bikes are 50% to 75% higher than comparable bikes a year ago. I think you may wind up overpaying and settling. By settling I mean selecting a bike because it is available. Maybe not the best fit, or the correct type(road, gravel, hybrid) but available. If you can convince yourself that riding an older heavy bike is conditioning you for longer rides, similar to people who run with a backpack or ankle weights. One of the downsides of inexpensive heavy bike is the riders get discouraged and loose the drive to push through discomfort. You seem to have gotten past that. Perhaps instead of setting a mileage goal set a time/distance goal on some rides. That way you won't be mentally locked in the distance mindset.

The damper and tyres are constantly draining on your energy, and the heavyness kills you uphill. Your bike is just murderous imo. ;)

You only need this downhill stuff with big tyres and damped frame to go downhill on rough tracks, fast. I wouldn´t give a damn about my downhill speed, you have many miles on the road to catch up. Plus you can leave your integral helmet and protectors at home.

Uphill on bad tracks, you don´t need the big tyres, and 10% less weight (you+bike) is a blessing if you go uphill on- or offroad.

If you don´t want to buy one of those fancy new "gravel bikes", I´d get any old touring bike, take off the mud protectors, get a normal road front tyre, and a slightly larger one in the back, so you retain traction on a few bad spots uphill.

Add 5-8kg to your bike*, go for a ride.

Take it off, ride again.

Expect the difference in buying a lighter bike to be at least as good.

^{*Try to add mass uniformly, so as not to affect balance.}

Unless the trails are quite rough, front suspension is probably making you work harder even there. They'd have to be properly technical MTB trails for rear suspension to speed you up. And on the road suspension is always going to hold you back (if you can lock it, it's only a weight penalty).

Some slightly relevant tests I've done recently by riding the same trails on different bikes/tyres may shed some light:

On fairly smooth dirt and packed fine gravel my tourer (with 28 and 32 mm slick tyres) is faster than my 29er hardtail. I have to dodge a few things but can keep the speed up, and sometimes have to stand up on the pedals On loose coarse gravel (a farm track with some ruts and holes but far from technical) the tourer beats the MTB with knobbly tyres like the OP's but changing the MTB tyres to WTB Nano 2.1s made that quickest (suspension unlocked). Both bikes weigh about the same, and the coarse gravel is slightly uphill. The difference in any case is marginal, at least in fairly dry conditions.

On the road, my tourer and my hybrid (MTB-like riding position, though narrower, a little heavier than either with its accessories) are both faster than the MTB. The hybrid has slightly grippy commuting tyres.

It depends on how rough the trail is, and the ratio of trail to road, but it sounds like a bike optimised for road but capable of some dirt would be much easier. That could be called a gravel bike, cyclo-cross bike, or adventure road bike with drop bars, but a light hybrid would also work. Tyres with just enough grip that you're safe and can keep moving forwards should be optimal. I'd expect these to have an almost smooth centreline but tread towards the shoulders for more grip cornering and on soft surfaces. A little slipping in occasional muddy patches is tolerable but requires more handling skills, so with practice your choice of tyres may change - or indeed with the seasons

I think the question of "worth" depends largely on what you plan to do after you achieve your 40-mile goal. Is that as far as you ever want to go, and all subsequent rides will be shorter? Or will you set a new goal of going farther, faster, and/or higher?

It all depends on how serious you are about riding. 20kg is a really heavy bike, and you will feel absolutely liberated and inspired when you ride a 10kg bike. But if you are a casual rider doing casual rides with limited (or no) goals, then your heavy bike is probably all you need. If you are going to be a serious rider doing more serious rides, then I would definitely invest in a better bike more in the 10kg range, of which there are many.

If you are lacking energy, what you need is more energy.

From reading the comments, you say you are doing 33.5 miles over a four hour period and not eating. Most people would want to eat something in that time if they were sitting on the sofa.

You must eat. (I assume you are hydrating)

Taking on something every 60-90 minutes should see you to your target without too much trouble. You can buy specialist energy bars, but you don't have to - jam sandwiches were good enough for Graeme Obree - I favour something that is both sweet and savoury such as honey roast nuts or peanut butter and honey sandwiches. Either way you probably want something that will provide a kick of short term energy, and more gradual release long term energy.

As per other answers, there are gains to be had from modifying your bike or getting a new one - whether they are worth it depends on how much money you have, how much more cycling you plan to do and how important your goal is to you. But these gains will not see you through a 4-5 hour ride without eating. And, I think, if you eat, you can probably do your 40 mile trip on your current bike on your next ride.

In terms of making a ride of about 40 miles one big thing to consider is are you constantly pedaling at an "optimal" speed. From the below article an optimal cadence is 60-90rpm. This can be maintained even on hills by changing gears appropriately.

A bike computer with a cadence sensor will help with keeping an even cadence.

reference: https://blog.wahoofitness.com/cycling-cadence-what-is-it-how-to-improve-yours/

The only problem is, the bike is very old and weighs just shy of 20 kg. It's a heavy one! It also has Kenda Kinetics tyres, I think they are made mainly for off road so not the fastest.

It is indeed quite heavy. In this very special case, aiming for ligher weight in moderation is useful. However, many lightweight bikes are not durable, so one needs to carefully estimate whether the weight loss is made through improved technology or through reduced durability.

To fix the tires issue, choose Continental Grand Prix 5000 tires, preferably in 32mm width. They are about the best rolling tires one can find.

My main goal with cycling is to cycle 40 miles to a target destination I have. So far I have gotten to the point where doing 20-22 miles is pretty doable for me now. My best is 33.5 miles, after doing that one I was really lacking energy towards the end. I live in a place where "High" is in the name, it's full of hills and coming back to those hills after doing that kind of mileage is killer, especially on my 20 kg bike, towards the end I was feeling like the 40 mile is not possible, but I had the same feeling when first doing my 20 mile runs.

You can realistically lose about 5 kilograms from the bike weight without reducing durability. A reasonable equipped bike weighs around 15 kg. If you remove rack, kickstand, mudguards, and lights and use regular front hub instead of hub dynamo, something like 12 kg might be doable without doing it at the expense of durability.

So the 5 kg of achievable weight loss means you reduce the system weight from 90 kg (70 kg rider, 20 kg bike) to 85 kg (70 kg rider, 15 kg bike). That's only 5.6% weight loss. And it only matters when riding up hills. So a hill where you could achieve 13 km/h, means you can now achieve 13.8 km/h. The difference gets lost in the noise. If you are a heavier rider than 70 kg, the proportional weight loss is even lower than that.

Will I notice enough of a difference, given that I can now do 20 miles with ease on my heavy bike? Or do you think I should just suck it up and try to reach my goal with my heavy bike. Am I just being an unreasonable amateur blaming the heavy bike?

The difference of 0.8 km/h in uphills most likely gets lost in the noise. It is barely perceptible in your uphill speed, but most of the time you are not riding uphill, so the final average speed will not be much different.

In my opinion, the loss of 5 kg from bike weight is worth about 1000 euros if you ride a lot. It might be tricky to find a 15 kg bike for 1000 euros unless the bike style you prefer is one of the popular styles. For example, I prefer touring bikes and those usually cost around 1500 euros, because they are not in fashion currently.

If riding a bike is not something you are planning to do for the next 10 years continuously, forget the weight loss upgrade.

My estimate is that if you are not using Continental Grand Prix 5000 tires already, purchasing those will cost only a very small amount of money but will make more perceptible difference in your average speed than 5 kg weight loss from the bike or from the rider.

For anyone with a heavy bike, upgrading to a lighter bike is definitely worth it, especially if you live in a hilly area like me!

I disagree with this conclusion. Light weight, if done at the expense of durability, is a bad feature.

My rule in selecting bicycle components is to first estimate the robustness and only after that choose light weight, if the weight makes financial sense. In bikes other than folding bikes, I usually value one kilogram of weight reduction, if not done at the expense of durability, at approximately 200 euros per lost kilogram.

This 200 euros / kg rule applies only to parts that last practically forever. For example, for something like a chain, it would be foolish to reduce weight. A chain lasts only few thousand kilometers. In a chain, losing kilogram is worth only 12 euros and a chain only weighs about 0.3 kg, so a hypothetical chain with zero mass (unrealistic) would be worth only 3.6 euros more than a regular weight chain.