Does the student even know it's acceptable to make grammar mistakes in the interest of efficiently explaining their ideas? Literally if the instructor has never taught this then the student has no chance of knowing this. It may be worth observing that both in "the real world" of industry and later on in academia the majority of communication is naturally done hastily and under time pressure, in the form of a stack of email that must be resolved before one can get on with work. And email, of course, is influenced by the speaker's own conversation style, and it is usually clear enough.
ESL speakers can become fluent while making more or less systematic categories of grammatical deviations that are more or less within the bounds of business English. But the entire culture of written examination in school emphasizes the wrongness of the deviations and not the fact that the communication itself is clear and fluent, and this may be the main issue with your student's struggles.
A certain amount of grammar mistakes only modestly impacts readability. Even if it moderately impacts readability, not penalizing these mistakes in a time-pressure quiz or test format is a good idea. This applies to all students. ESL and all students deserve time offline to polish.
Work with your student to find compromises between what they can express fluently and efficiently, and what they can express grammatically in the allocated time. The student may understand this as a bargain: they focus on clarity, and you forgive mistakes. I would suggest being collaborative with the student. Perhaps you will have a need to penalize some mistakes that compromise clarity, but the student deserves the chance to work through which mistakes are better to make.
I want to emphasize an important point about this answer: I advocate no special treatment in grading for the ESL student here. Education should be specialized to the students in need, and changes to evaluation is a last resort. (Of course reflecting on this situation may lead to new perspective in grading, but that's the point of asking questions like this in the first place.)
Now, the OP's question can be answered independent of the act of plagiarism, but I do have something to say about that regarding the circumstance: I would emphasize that inasmuch as justice is about punishment, it is also about when punishment ends so that the perpetrator may get on with their time in society. In this case, it means not depriving the student of their right to quality education assuming the punishment is anything less than expulsion. At the risk of being political, I might suggest that part of what turned the student to this crime was lacking any other way to succeed. That is within your ability to impact as an educator, and relating this empathy to a path forward in this student's education has nothing to do with lessening or excusing the offense.