IF a 13-23 was the factory cassette, I'm pretty sure you have a short cage RDER and it won't handle a larger cassette.

Cheap route- Get a low level Shimano RDER such as this-
https://www.amazon.com/Shimano-Acera...LwhvD_BwE&th=1
And a cassette with a 34T largest cog.
You most likely will have to lengthen the chain, so it may be easier to just get a new one and shorten to fit if needed.

A 9 speed offers more options, but would also require a 9 speed shifter (s?) and chain.
That's often a good option if you HAVE to replace the R. shifter.

EDIT-
IF this is your bike, you should be good up to a 28T cog.
https://www.bikepedia.com/Quickbike/...%20&model=R600