What should a supervisor do about a PhD student who does not try to improve? What should a faculty supervisor do about a PhD student who does not seem to be trying to make their work better?  Let's assume the supervisor has given detailed and specific instructions, which the student agrees to follow.  But the student either does not finish the instructions or includes many errors.  The supervisor points out the errors.  This process has been repeated for several years, affecting both the student's research and job search.  The student is from a different country, and has not adjusted to academic communication practices, despite being aware of the need to do so.  The student has done a lot of things, but continues to have poor quality execution and pace.

Typically the supervisor does not wish to kick students out of the group.  What is the right course of action?
Dealing with wayward students comes with the territory of being an academic. Not every student will be successful and, unfortunately, you have to be prepared to deal with that.

It sounds like the situation has gotten out of hand, as it has gone on for years. Expectations need to be set early and often, so that, if things do go off track, the problems are addressed as soon as possible.

The other answers offer good suggestions, but your comments suggest that you [I'm assuming you are the student's supervisor] are worried about how it affects you. So I will offer a different suggestion:

Involve other (more senior) faculty from your department.

If you involve other faculty, they will share the responsibility for the outcome (thereby reducing the possible damage to your reputation) and, more importantly, they will be able to give you better advice than random people on the internet.

Where I did my PhD, there were established procedures to involve other faculty: Early in the PhD, each student had to pass an oral qualifying exam involving several faculty, which helped identify struggling students and provide feedback. Also, there was an annual student review in which the entire department met to discuss the PhD students one-by-one, which allowed their supervisors to raise issues and get input from others. Finally, each student's dissertation committee was officially a standing committee (although for most students it only convened for the defense). That is, for problematic students, the dissertation committee would convene regularly to assess progress and set goals for the student. The committee thus shared responsibility for "kicking out" students.

Does your department have similar procedures? If so, follow them. If not, suggest that such procedures be implemented and informally approach other faculty in your department for help.

Remember that it is in no one's interest for a student to continue on a bad trajectory (including the student's).