To put the fact that it's a fourth email aside for a second and answer the question
Is it okay to start mail with “I apologize for the additional email,
but…”
Even if it were the second email, my general policy is don't apologise needlessly.
It loses its meaning
I want my apology to mean something. If I mess up and have reason to sincerely apologise, then I want that to have weight behind it. Over the past few years, the number of times I've used the phrase "I'm sorry" or "I apologise" in an email is probably < 20 times. I want my boss and coworkers to know that I'm sincere. If I use it all the time in cases like this, it loses its meaning.
It makes you sound like you lack confidence
Have confidence in what you do! If you send an email, be confident in what you say. Have purpose behind your words. Saying I'm sorry is like saying "I'm not sure if sending this email is the right thing to do" (which is exactly what you're asking). Either send it and mean it, or don't send it. Don't admit you don't know what you're doing.
If you do good work and have confidence in yourself, then people will have confidence in you and you'll gain respect from others. How do you lose the respect of others? (1) do bad work or (2) second guess yourself.
It's just extra reading for the recipient
A busy recipient won't want to read through sentences of fluff before getting to the point. Get to the point concisely and keep your emails short.
So while the accepted answer is correct, here's an elaboration on why you shouldn't apologise in this case; save it for when you really are sincerely sorry.