[dogsbody70]

we often say "HANG ON" when someone is waiting for us. If they are in a hurry and we want to finish what we are doing-- such as finishing off an essay, we might say: " Hang on a minute- won't be long" meaning we want them to wait for a short time until we have finished what we are doing, So in that context we are asking them to wait a minute or two for us. "Hang on, I want to look in this shop." expecting a friend to wait a minute while they look in the certain shop.

I would not say:
If I hang on to continue this thread, I would become a Master of English!.
If I persevere with this thread Yes if you persevere you will make good progress-- but Hang On is incorrect in this context.

Maybe someone might say "Hang on in there" if you are feeling everything is too much for you-- we might say " Hang on in there, don't give up, you can do it--" sort of thing.