If I had an affair, resulting in a child can I fight for custody of that child?

You aren't going to get much of an answer unless you give us more information. Are you the mom or dad ? Would you have to 'fight' for custody, would the situation be dire enough for you to have to do that? The answers are different if you are the birth mother, or the biological father. That's why I pointed out you not mentioning your gender.

Tick

Tickle did not realise I am a mind reader,

So let me guess and see how close I get, you are the father and they have now filed child support papers on you.

In that case you can file for visits, and if you live close enough even joint custody. The only way to get full custody would be to prove the mother is a danger to the child and completely unfit.

Yes that's the case and she is unfit (lives about 200 miles away) she's already tried to kill herself a couple of times, her mam has to take the child most nights cause she can't cope! What's my chances?

If you can prove her unfit, then you go to court for custody. Your chances depend on your proof of unfitness. Get an attorney who knows local courts.