Not sure what model you've got (that info would be helpful), but the battery isn't generally part of the airflow (any ventilation is usually just to keep the battery cool).
If you've got an automatic battery, there's probably a hole into it so that your suction can activate it, but it wouldn't affect airflow AFAIK. The place to look is the atomizer and the cart. If the atty is flooded (too much liquid down at the bottom), that affects airflow, and so can getting liquid in the air duct(s) in the cart (that also happens when you flood the atty -- liquid comes back and gets in the side of the cart where air goes
through).
Assuming you're not using a "cartomizer" with cart and atty in one unit, try taking the cart off and blowing
through the end where you suck, and then taking off the atty and putting the battery end in your mouth (put your mouth over the outside part, not the brass stem) and blowing hard for a few seconds. In both cases, you might want to have a paper towel or something to catch any liquid that comes out.
(If none of this makes
sense, try posting the model of e-cig you're using, and someone familiar with that model may be able to help.)