Thanks for your question.
In English we have highly idiomatic phrases called phrasal verbs.
Phrasal verbs consist of a verb and usually a part of speech that is never changed
Let's look at a few examples.
1) I finished all my homework. (The task is completed.)
2) I'll finish up doing the laundry. (I will complete, entirely, washing the clothes.)
She'll finish up her degree in chemistry later this year.
(She will complete all her schoolwork to obtain her degree, entirely.)
Notice, the nuance, "To finish up" means to complete something 100%.
If you only "finish", that doesn't mean the task is fully completed.
3) Finish up eating everything on that plate! (Eat all of your food.)
But if I say, "Finish your dinner, we got to go."
(Finish and eat whatever you are going to eat for "now" because we are leaving.)
I will let someone else answer to "finish with" for you.