Definitely do NOT tell them. You will be looking over your shoulder all the time at work being paranoid that you'll bump into one of the nurses or doctors you meet at the clinic - I told the first clinic I went to and ended up on a placement for six months in that hospital
Before my escorting I used to be in an open relationship where I knew my boyfriend was having sex elsewhere, which I was fine with cos I was on depression medication that lowered my sex drive, and I understood he had 'needs', so that's what I say when I go to the clinic. It's good as a sex worker to get throat swabs (if you offer owo), and some clinics won't routinely do this, unless you tell them you're a sex worker, so I use the boyfriend sleeping around excuse and say that I give him unprotected oral, so ask for a throat swab to be 'extra safe', and ask for an anal swab too if you offer anal.x