The Simple Loop Conjecture is as follows.
Every non-injective map from a surface group to a 3-manifold group kills a simple closed curve.
As there are all sorts of 3-manifolds with abelianisation of rank two, I think the answer to your question is unknown.
UPDATE: Sorry, I wrote the above too hastily. I should have said 'I think that the kernel is not known to contain such a loop.' On the other hand, there may well be examples of such maps with no simple loops in the kernel. You could try looking at Louder's recent preprint on the Simple loop conjecture for limit groups, for instance.