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My apple trees are in their 3rd year and not growing very well. I think this may be due to grass infestation around the mulch and I suspect I'll have to dig down a few inches to get it all out. Is this safe to do? Will I be damaging the tree if I do so?

I would be very careful if you start to dig close to your tree Decado.
Fruit trees are relatively shallow rooted with the majority of the feeding roots spread out in a fan just a few inches below soil level.
So if you do not want to kill the grass with a weedkiller then try and remove it by digging down as shallow as possible.
Once you have removed it get a good layer of mulch down on the bare soil to encourage the roots to get going.
The main problem with grass around young trees is that it does use a large amount of water, so in their first few years they do need regular watering to get them going.
Interestingly though.....when fruit trees are grown in bare soil strips, as in commercial orchards,,,they do seem to require more phosphate added to the fertiliser than those growing in grass.

Hmm... I would just rake off the mulch, mark off the area by edge cutting the grass along the perimeter BUT NOT DIGGING/LIFTING THE GRASS (or maybe remove just a 4" wide band all the way around), put down some unfinished compost (nothing recognizable but chunky and still fair amount of earthworms present), water well, then wet cardboard the area and mulch with whatever I can enough to cover and hide the cardboard, then decorative mulch if I'm so inclined with some kind of uniform mulch. Grass/sod should die and break down underneath the cardboard.

...just yesterday watched my neighbor's kid and neighbor laboriously digging out the grass/sod around their front yard tree. The teenager was REALLY digging into the ground and I was thinking yikes Maple tree is shallow rooted.... Then they wheelbarrowed all that rich sod off to the woods, THEN spread bags of purchased tops soil and mulch. For all intents and purposes, it looked very nice afterwards.