Right now y'all are using the three slots allowed for a site by default - it's not a matter of them being stuck, that's just the default for everyone. We can expand the slots to four or five - five is the max - but we tend to do so thoughtfully to ensure that there's a strong need for it.
Many of the requests we get for this come from sites that have used the three slots but one of the close reasons they have active currently gets very little usage and so we recommend that they retire an unused reason rather than having a fourth one. This is important because the close UI can already be somewhat confusing and muddying the system with superfluous close reasons exacerbates that issue. We're doing a lot to improve that experience so hopefully that won't be as big of an issue moving forward.
The first thing that I do when I get a request like this is to review the 10k user tools that show the close stats for the site over the last period of time if you have 10k reputation here, you can view the last 90 days by clicking this link. If you don't have 10k, I'll show you what it looks like here:

Here you can see the usage of the different close reasons the big ones in the first five rows are the ones on the "community specific" page of reasons, so the three custom ones for EE along with the catch-all "other" and the migration option (belongs on another site). Of the three custom ones, your repair reason is used the least, only used on 5.23% of questions overall - now, this is a small percentage but since y'all close quite a few questions, it's still 70 questions, or about one per day. The other two are nearly triple that, around 13.5% each.
Considering this, I'm willing to increase the number of slots to four but I'd really like y'all to think about your close reason more carefully - not because I think the reason you want to add is bad - you're already using it as the most-created reason for the "other" option:

There were 70 questions over the last 90 days closed with some variation of the homework close reason and - well, I'm supporting making this a close reason for the minimal reason that some of the messages that have been created weren't necessarily very kind:
I’m voting to close this question because this is homework, dumped on us, without any own effort and without even asking a question of one's own understanding.
I’m voting to close this question because stack exchange isn't a homework solved on demand service.
I understand that it can be frustrating to see these questions over and over and even to have a request like this one in place to get it made an official reason makes you hopeful and looking forward to a time you don't have to recreate the text over and over.
So, what I'm asking is that y'all come up with a few things - close reasons are much more complex now than they were when your last reason was added. Now, rather than only needing one piece of text that did all the work, you can now target the text to different audiences. So, rather than one, you need five elements:
- Brief description - this is the Bold part of the close reason that appears in the close vote UI when closers are voting to close the post. It's the equivalent of "Needs more focus". For this, maybe something like "Homework"?
- Usage guidance - this tells close voters when to use this close reason. So, you could put something like:
This question is about a homework problem but is missing additional details such as the asker's attempt at a solution and an explanation of what they do not understand.
- Post notice close description - visible to all users. This is a general note about why the question was closed. It can include links to resources that explain the site's policy. An example might be:
This question was closed because it is about a homework problem but doesn't provide sufficient information to be answered. It is not currently accepting answers.
- Post owner guidance - this additional information appears in the post notice but only for the asker of the question. It should contained detailed information about how they can improve their post and may also include links to help here on meta or in the help center. For example:
While we will not provide the answers to your homework, we can help you understand how to get to the correct answer. To do this, we need you to edit the question to include more details including what work you have already done and the solution you came up with and specific information about where you are stuck on solving this problem.
- Privileged user guidance - this additional information appears in the post notice but only for users with the close/reopen privilege. It is designed to help them know how to guide the asker in improving their question or inform them when the question should be reopened. You might say something like:
Please guide the user with specifics on how they can improve their homework question - if some of the details are in the comments, consider editing the question to include them. If the necessary information is added to the question, please vote to reopen it.
Y'all have a ton more context than I do, so I assume you can do a better job of crafting these elements than I can but I do hope that this gives you some help in how to do this. You have 500 characters to work with for the last four of these. The first is much shorter (100 characters). Please let me know if you have any questions. Once you've completed this, ping me and I'll get a developer to increase the number of close reasons for you.
Additionally, if y'all would like to follow this format for your existing close reasons, please do drafts for them here on EE Meta and a CM can edit the existing reasons to follow this pattern (have a mod tag the discussion status-review once you've settled on the text). Since moderators can not edit the close reasons, you'll need a CM to help with that but we're happy to do it if you wish.