Is it just a nowadays English adaptation of the verb to push ?
German has two homonym verbs puschen:
1. puschen = to pee
There is an old and almost forgotten German verb puschen that means to pee (to urinate). It is very rare, and as already said, old and almost forgotten, and it's only alive in a few regions as part of colloquial speech (sorry, don't know which regions).
2. puschen = tu push
This is a relatively new and well known loanword, derived from the also very well known foreign word pushen which is a copy of the English verb to push, which itself is a French loanword (pousser) with a latin root (pulsāre).
The verb puschen does not exist in German. Do you mean the verb pushen? That's a 100% cognate of English to push, along with the slightly sharper pronounciation of sh in contrast to sch. It has been taken from English not before 40 years ago, I think.
Or do you mean the noun Puschen (pronounced with a long u)? That means house slippers and is mostly used in the phrase Komm in die Puschen! (Hurry up!), and as a synonym for shoes in general.
Or do you mean the verb pfuschen? That means to botch and is pretty easy to distiguish from pushen by the explosive f == pf sound on the beginning.