Kittie: Your tree looks a little overly vigorous in the top. But First cut out any limbs that are in your way, crossing, or otherwise out of place. This would include the branch you mentioned. Either cut them out completely or cut them back to a less vigorous lateral shoot that is headed in the direction you want a limb to be.

Then cut back the most vigorous shoots in the top of the tree to less vigorous side shoots or remove some of them entirely. Do all of this cutting in August only if you live in CA. If you live elsewhere, I would do it after you harvest this yrs crop. In CA you can get serious disease problems by pruning apricot any time but late summer.

Each yr after this continue to cut out the most vigorous shoots in the top of the tree each summer.

Basically the first rule in pruning a tree like this is to cut out any branch you don't like and do it every yr. If you do that the tree will not get too big and you won't be removing too much at once.

The Fruitnut