I don't like your T's behavior either, but maybe he is just addicted to the computer! That's not good, but it happens to the best of us!
However, he should be able to control his behavior when he is with a client. He's always sounded like the perfect T, but no one is perfect, so maybe this is his one fault. Or, maybe he's waiting for a particular email from someone? Or has it been going on too long for that?

If it were me, I'd tell him how much it bothers me until he gives me a satisfactory answer. It's not professional behavior. Why can't he check his email between clients? I'm sorry you have to deal with this, sunny. It's hard enough for you because you don't see him so often. I hope you can work it out.