How to minimise student's discomfort when suggesting they need to improve his/her English?
One way to approach this is from the perspective of why the quality of prose matters in the first place.  After all, why should we bother with spelling, grammar, etc?
Having good spelling, grammar, etc. is important because not having them puts roadblocks in the way of communicating with the reader.

You can minimize a student's embarrassment by emphasizing this aspect.  Assuming that student is otherwise doing well, you might can bring up grammar by approaching it from this angle.  You can emphasize the pragmatic aspect that, in pretty much any career they choose, written communication will be important, and that it's important to not have others' appreciation of their good work be impeded by problems with grammar and spelling.  It's really the linguistic equivalent of keeping yourself clean and well-groomed.

If a student is not doing well with the core material... well, then grammar's not really the key problem to talk to them about anyway...