Please, help! When I give the remote control, the participant can't click.

You have not stated what device the host is using and what device the student is using. IOS devices have additional steps for the student to follow. First check https://support.zoom.com/hc/en/article?id=zm_kb&sysparm_article=KB0065790 then be sure IOS iPad students tap the icon to access remote control. Note that iPhone students do not have remote control and will not show up as a possible device.

I am using MacBook Air M1 and my students are using either MacBooks or Windows PCs or laptops. The problem is that sometimes the remote control is working excellent, but then abruptly stops working at all. A student takes the remote control but can't click on anything at all. 

E.g. I had 6 lessons yesterday and on the first three lessons the remote control was working, on the fourth a student couldn't take the remote control (although she has lessons with the other teacher in Zoom and took the remote control in the lessons). After the lesson I asked my husband to check if he can take the remote control in my conversation ( before that we checked and he could) and he faced the same issue. So the problem is definitely in my computer and my Zoom program.

Were you able to solve this issue?  I am having this issue now.