Q:

What to do about students who ask for help too often?

For my writing courses, about 5% of students will come to me prior to deadlines asking for help with their paper. I see no problem advising students, as I often similarly came for help when I was an undergraduate. Recently, though, I found an increase in students who apparently just want to abuse this:

Students will bring me some plagiarized work, showing it to me early, as a sort of test if I will notice. It seems difficult to punish plagiarism when the paper is not yet submitted.
Students will bring in papers again and again, with little changes put in at each stage, hoping their minimal effort each time will be sufficient to reach their goal of a "D".

I've tried stopping students, but then they are angry when they see the "F" that they hoped I would help them get away from. While most of these students are probably just incredibly lazy, there is a chance that some among them are genuinely trying to improve, but just struggling a great deal, and I can't see it.
How might I go about blocking such abuses?

A:

Here is one strategy:

Specify the level of changes the student needs to make before they can come back again.  
Request that the student shows you what changes have been made by comparing the old and new versions of the assignment.  
If student has failed to reach the level suggested or fails to show you the differences, simply tell them to go away until they have made the requested changes. Simply state that your previous comments still apply.

Adding a time delay between when they ask and when you give feedback or when you give feedback and when they can come again might also help.

A:

While there are some good answers here, I will just add a few thoughts from my own experiences.
I also have plenty of students who try for minimal work just to get a pass. I used to explain to them what their grade would be and why but in the end, all they heard was what their grade would be. If it was pass they stopped listening. Of course, this is quite unhealthy for their longer-term success.
I have since changed to not telling them what grade I would give them before they submit, partially because of JeffE's comment to this question. Now, I focus ONLY on showing the students how to judge their own papers. I explain that I will mark when they submit but if they want to understand how to mark their own papers, I will help them.
I do not check for plagiarism on drafts (though some students do ask me to). If I see something which looks like plagiarism, like the level of English goes from very poor to perfect, I do let the students know that this looks like plagiarism and, if it is, they should fix it with proper citation before submitting.
In short, focus on teaching them how to grade their own papers. This helps a lot if you provide a rubric. Do not tell them "I will give you a  'D' if you submit this." Instead, tell them "Tell me what you think this paper deserves and explain to me why you believe that." Then help them develop that skill.
They should become independent learners, even if they don't want to.

A:

Another strategy is to provide more guidance about what sort of help you'll provide and how they should ask for it. Possibilities:

The first time you bring your paper to my office [or during specific dates], we will focus only on whether you have answered the question.
I will be happy to discuss aspects X and Y, but never Z.
Bring the rubric with you. Be prepared to explain which part of the
rubric you most need to address in your draft, and why. We'll focus on that part   of the rubric during our meeting.
Before you bring your paper to me, prepare a list of specific questions you would like me to answer. Bring two copies of the list, one for each of us.
Before you bring your paper to me, show it to someone at the writing center. Bring with you your notes from that visit and the draft you showed to them, along with the draft where you implemented those suggestions.
Always bring two copies of your draft--one for each of us. Be prepared to write  detailed notes on your copy. (Then you can keep your copy and you'll be able to bring it out the next time and say--ok, what did you change? Or you can compare it to the draft they turn in to see if they made significant changes. And if they plagiarised it, you'll have a record.)
Each time you show me your paper, I will expect you to create a checklist of things you will address. The next visit, you need to demonstrate that you have completed those things by bringing a draft with changes marked and annotated.

Another thing you can do is assign "draft" deadlines as well as final deadlines for everyone, then give the feedback you think is important on the drafts (maybe using a rubric) and not meet incessantly with a few students.
Another thing you can do--particularly with very structured papers that you have assigned before--is provide to the class a list of the common problems students have. Then when a student shows you a draft, you can initiate a discussion about which common problem the student thinks it illustrates and what can the student do to address it?
Another thing you can do is limit the number of times you will look at a draft, or the period of time during which you will look at a draft. 
I have had a similar problem with students wanting me to grade work before they turn it in for a grade--not just on writing, but with all kinds of assignments. Sometimes they are so lost they are completely stuck. Other times, they seem to be trying to minimize their workload by increasing mine. ("Just tell me in exacting  detail what to do, I'll do that and no more, you'll give me an A.") But what I want is for them to learn how to assess their own writing! 
Whatever strategy you use, when you meet with students, try to elicit their comments. If you give a suggestion, and they nod, then ask them how they expect to apply it. What changes will they make to a particular sentence or paragraph? Are there any other places in the text they should also make that change to--which ones, and why? If they are to provide more evidence, ask them where they intend to search and how. Then you can better assess whether they understand what you've told them and whether they can actually do it.
I often ask students about how they produced the draft. Which parts were the easiest and which parts were the hardest, and why? Which parts do they like best? Which parts do they think need the most revision? Sometimes I ask them to imagine a reader--if they were to show their paper to (mom, roommate, employer, etc.), which part would be the most controversial? What would need the most explanation? If their reader disagrees, what would that person likely argue back? The more they talk about their writing, the more opportunity you have to say, "Yes! What you just said--write that down. Now do that!" 
Another benefit of getting students to talk about their draft is that it is tough to talk about something you didn't write. If they let you know they copied passages, you can remind them (or explain how) to cite them properly. If they pretend they wrote every word, you have a different problem to address. 

