What shows in the hamburger menu is not controllable by a member.
But it can be based on Group membership.
For example, if I am a member of the lolcat group, and there is a lolcat category, that category can have permissions set so that only members of the lolcat group can see it.
I would see it in the menu.
If I wasn’t a member of the lolcat group, I would not see it in the menu