Making bicycle semi-motor powered

ebay.com, "Electric Bike" brings up dozens of possibilities. "Petrol bicycle" brings up several petrol option. Any of these kits will do the job, but you will get what you pay for.

Electric kits are expensive, at least twice what you have suggested.

In the price range you are looking, a petrol motor version might be best. It's tried and old technlogy, is reliable, is cheaper and has a longer range than electric. Electric does carry the "Green Tax", is new and relatively expensive for good solutions.

Be aware of local laws as far as limits to motor power and speed go. Most locations limit the power to about 300W and often speeds to below 20km/h or 25km/h (above these the motor is not allowed to provide assistance).

What research have you done, is there any reason any of these is unsuitable?

Having watched the kids next door attach one of the petrol kits to a bike over a few weekends and try to make it work, I suspect the cheap kits don't work very well. They spent a lot of time trying to make the motor run for more than a few seconds. They're also not going to be legal in the EU or some other places because they don't meet the emission regulations or other rules. In Australia they're not legal because the power assist does not cut out at 25kph.

The very cheapest electric kits on eBay start at about your maximum price limit once you add shipping and any extra parts you need. This Instructables suggests twice your price is a reasonable starting point. If you're in the USA then bicycle power has a huge list of links (the US has few rules so there are a lot of options). Those are likely to work only after a lot of effort and expenditure on your part, if they can be made to work at all. My local ebike shop used to sell kits but stopped because they're so failure prone, even the good ones. It's actually cheaper to buy a proper ebike than to attach a kit to a bike then pay for all the things you need to make it work.

To get the price you want I think you will have to DIY something. There are a few reasonable articles on the web. Julian Edgar does it the traditional way, WISIL have some articles on using radio control model parts to get a light, cheap boost motor. Either of those will eventually get you something that works, but will require a lot of work from you (as is often the case, you can sustitute your on labour for money).

Not that cheap, but there are a few companies making retrofit wheels that will add pedal assist to your bike. They will require very little modification to the bike. I'm not sure the cost, but likely at least $600USD to $800USD per wheel, if not more. Both appear to be pre-production, but open for pre-orders.

See the copenhagen wheel and FlyKly Smartwheel