Should I push for my paper to be published as an undergraduate? I've been doing a research attachment as an undergraduate in a malaria culture lab and my project involves the screening and hit prioritisation of anti-cancer drugs for repurposing use against malaria. Since the attachment is an official programme under my school, one of the deliverables is a paper at the end of the attachment and I was keen on writing a paper to be published in a journal(if it's accepted!)

I mentioned this to my mentor and he said it might not be worth publishing the paper or something of the sort but didn't elaborate on it much. Should I look into it and ask if it can be published in an undergraduate journal or should I let it go?
Perhaps your advisor is worried about the depth or novelty of your work, suggesting that it might not be worthy of publication. s/he might be correct or not. But there is no reason to restrict yourself to an undergraduate journal. If the work has the right quality it can be published anywhere that the referees and editor will accept it. The fact that you are an undergraduate has no bearing on the issue of quality. 

You can submit a paper, certainly, though whether as a sole or joint-author depends a bit on your field and location. It might be worth it just for the experience. 

But speak again with your advisor/mentor to find out why they were originally a bit dismissive. And look at what appears in any journal you might submit to in order to judge your paper against what they normally accept.