It is a little abbreviated. I think the full sentence might be something like: "I want to get a bit of work done". 'a bit of' means 'a little' or 'some'. In other words, the speaker wants to complete some work over the weekend.

"Get a bit of work done." means you are going to do a little of the work you need to do, but you are not going to do all of it. People use this when they want to do enough work that they feel like they have accomplished enough to allow them to stop, to not feel like they wasted the day, or to justify doing something else more fun. It doesn't have to always be "work.: For example, "I got a bit of laundry done this morning, so I treated myself to a trip to Starbucks." Or "I got a bit of studying done this afternoon, so now I'm going to meet my friends for dinner."

As I have been studying Spanish, I have learned to appreciate these all purpose words we have in English that may not exist in other languages. One of those "all purpose" words is "get". So, my understanding of the sentence "Get a bit of work done" is that it means "Get (accomplish) a bit (a small amount) of work done (with Get...done means accomplish). Movies are not known to have a high level of English as it sounds stuffy and pretentious where slangy every-day language is more relatable.

"Get a bit of work done." This means that the person will be doing a task/work/activity, but he/she will not do all or most of it.

Get a bit of work done. > To do some minimal amount of work so you don't feel like a lazy person. It's basically a polite way to say that you want to work so you don't feel worthless/like you haven't accomplished anything.