How to deal with a research internship manager with very bad English?
It is the professor's responsibility to make sure that the instructions you are given are clear and understandable, and besides it's in their interest to make sure that you are able to do your job. So you should definitely talk to the professor about this issue, preferably giving a few examples and explaining clearly and objectively why it's a problem for your work. As you are clearly aware, it's not a matter of criticizing the person, it's matter of finding a way to work efficiently.

On a more diplomatic note, I assume that this PhD student has to write papers and will certainly have to write their PhD in English, right? So it's also crucial for them to improve their English level. Maybe you could suggest helping them at the same time: when their instructions or explanations are unclear, you would propose a better phrasing or point out why something is a mistake. The idea is to avoid that the PhD student feels bad about the whole thing, and this way they can appreciate that you're also doing some effort to make things work.