Leave the secondary growth or remove?

Most modern apple trees are grafted onto vigorous growing rootstock because the new varieties have weak growth. That is why the suckers are growing much faster than the tree. The knobbly lump at the base of the tree trunk, where the scion/graft is, should not be below or close to soil level. It is always dodgy strimming around the base of any grafted plant as they can so easily be damaged, once the bark has been broken suckers will almost always appear. Even catching it with a hand fork or trowel can cause a problem.
If you want to keep your tree dwarfed or low in height I would be inclined to shorten the main leader now by 2/3rds to restrain upward growth. Take it back to an outward facing forming bud.

 

I think digging into the turf will just create problems with root damage and therefore suckers. Maintaining a good, thick layer of light-excluding mulch will eventually kill off the turf anyway.
An easy way to add nitrogen would be a handful of granular lawn food (definitely without added mosskiller!) or dried blood mixed into the bark. You need something slow-release that will compensate for the depletion by decomposition of the bark, without encouraging soft, sappy growth of the tree itself.