I know it's a cheap PSU but is this really fine?
When it is within specs it is formally fine. What do the specs say? For my liking +260 mV aren't fine.
Will this work anyway for most chips
This depends on the chips, but many 5V should be able to survive this. You definitely lose headroom when it comes to the maximum rating of 5 V ICs.
or should I worry about it?
I would at least be careful. That voltage is, for example, at the upper limit, if not above, what a typical 7805 regulator would deliver.
Anything I can do?
Check the specs. If it is out of spec one option would be to return it.
If it is within spec, you could of course open it, see how they internally generate the voltage and, depending on how they do it, adjust or alter the circuit. You will lose your warranty.