Background check, forced to resign from bank

Hi, I was forced to resign from a management trainee position at a bank earlier in October, because I failed to complete initial training. After that I took up a job with a three month contract (for the money). The contract-job ended prematurely, just from 22/10 to 24/11; because the project ended way earlier than expected.

I am "sort of" friends with the guy who offered me the contract-job and he said he could be the one I put down for reference check. My concern is, if there's another background check later, how bad does my "resignation" in October look? And how bad is the premature termination?

I wanted to get another job at some bank again. Maybe that's just wishful thinking.

I am a local, fresh university graduate.