Are you sure it's not the carto, and not the tank? Put one on the Provari without the tank, cover the hole for the juice if pre-punched (otherwise disregard that point), and with a drip tip on try a draw from the carto. If you have the same problem, it's the carto connection that is the problem. The two little airholes in the connector of the cartomizer must be blocked for some reason on the Provari. There are airflow adaptors available for this problem, but I don't recall what vendor sells them.
If you do have good airflow with just using a carto alone, then it must be something with the tank. I'm not familiar with using tanks with the side fill hole, so perhaps someone else can provide some tips on that.