Can I use rechargable batteries in an external flash?

A 6VDC power supply would actually have to be quite massive (think laptop rather than phone power supply sized), big speedlites can draw several amperes when charging.

There are a few older flashes that, as per the documentation, deprecate the use of rechargeables (eg the Metz 45CT); this should not be an issue with more modern ones.

The most likely cause is contact problems - the rechargeables might be just a bit mechanically shorter than your alkalines, or their contacts might have oxidised (which will cause problems with a high amperage load) - try cleaning the contacts (on the battery and on the flash).

If this is not the case, either that flash is broken by design (that is why the alkalines also don't last worth anything, since - as mentioned in another answer - they quickly drop to 1.2V too), or actually designed for lithium AA cells (similar to, say, the Kodak Z8612IS camera, which does take standard alkaline or NiMH AAs but lasts maybe 40 shots on them) , or your copy has marginally defective electronics.

I use them all of the time in several different hot shoe mount flashes. The only difference I've noticed is that when I put in a set of fresh alkalines, the battery level indicator shows 'full'. When I put in a set of freshly charged NiMH batteries, the indicator does well to show three out of four bars. However, a set of fully charged NiMHs that show three bars will last longer than a set of alkalines that show a full four bars before use.

The Neewer NW670C is listed as capable of using Alkaline or NiMH batteries. I would start by trying a different set of NiMH batteries from a different source. If you have the capability to test actual battery capacity, my hunch is that your 2000 mAh batteries aren't really 2000 mAh. There are a LOT of cheap NiMH batteries in circulation that have vastly overstated capacities printed on them.

If anything, rechargables should be better than alkalines because they can deliver more peak power. But that is a problem too, because they could cause a fire. Cheap toys refuse to accept rechargables because it would be unsafe. How do they know ? The + cap of a rechargable is a bit wider than a regular battery, so with a bit of extra plastic around the + of the toy it won't make contact. That extra plastic is there for your safety.

I find NiMH batteries to be unreliable. If you haven't taken good care of them then they won't hold much of a charge and won't deliver much power. A good charger that can do repeated discharge-charge cycles can tell you about the remaining capacity of each cell. One bad cell in a set of 4 can make the whole pack useless. Always bring spares, and spares for your spares.

The older NiMH batteries have a high self discharge, you can't keep them in your bag for many months. The newer low-self-discharge ready-for-use types (Eneloop etc.) have a lower capacity. I have seen failures with both types, or maybe I just need to upgrade to a better charger.

Finally, the discharge curve of rechargable batteries is different from single-use batteries. They hold their voltage longer, but at the end it drops more abruptly. There is little or no warning that they are almost empty, it comes as a surprise. Again, bring spares, even keep a set of fresh alkalines in your bag.

Try getting NiMH batteries intended for high current usage. For example, get eneloop pro (capacity 2500mAh with 500 specified charge cycles) rather than eneloop (capacity sth like 2000mAh with 2000+ specified charge cycles). They will last fewer total shots over their lifetime, but will deliver more per charge and will recharge the flash much faster.

The clincher basically is whether the startup current of the charge circuitry will draw so much current that the voltage drops below the voltage where the logic circuitry can still operate reliably. If it drops out and, for safety reasons, disables the charge circuitry when dropping out, you have a hen-and-egg problem for startup.

Generally for devices with a flash (either cameras with built-in flash or standalone flashes) I use the higher-current eneloop pro.

If you don't want to hop on the brand hype of eneloop, at least try looking for batteries that have among listed use cases cameras/flashes and motor toys (which also take a bunch of current) rather than just alarm clocks, wireless handsets, and similar devices with low power demands.

To add another possibility to the existing answers: NiMH batteries tend to have a plastic wrapping rather than painted metal. Also there may be a tendency to slightly bulge as they age. I've had some that were slightly thicker than straight alkaline, and also minisculily thicker than some others. As a consequence they tend to slightly stick to the (cylindrical in my case) battery slots of one particular flash. That means that the springs on one side of the battery are not likely to overcome the static friction of the battery slot reliably, leaving the contact at the other side of the battery somewhat prone to not engage properly.

So a particular set of batteries I have is comparatively unlikely to work in that particular flash I have, by virtue of at least one battery not making contact in opposition to the spring with enough reliability. Those batteries are also a nuisance to get out of the flash again: you have to shake pretty heftily.

In the end, a purely mechanical problem. Of course there also is the possibility for inserting at least one battery the wrong way round. Happens once in a while to the smartest of person.