Congrats on having such a nice tree planted in a nice Sunny spot. If it were my tree, I would place a thick pile of woody mulch around the tree to make a 6 ft circle. I would regularly water the whole circle. The soil will get better and better and thus invite the roots to grow bigger and bigger. In late Fall I would prune all lower growth up to about 3-1/2 ft height so that a "wide" pc. of aluminum or galvanized conduit can encircle the lower trunk area while the conduit sits flat on the ground. This slick sleeve will help prevent possums, coons, tree rats, etc. from climbing up the tree to harvest your future fruit. Next, I would clip the tops of vertical branches at about 6 ft or so to cause the tree to grow more branches sideways/lower to eventually form a shorter umbrella shape. Because one day when someone is on the upper ladder steps and standing on tip-toes and reaching sideways as far as possible to hopefully pick a beautiful ripe berry that is calling out to you.....and the ladder starts to fall sideways....and so do you....Well, picking fruit from a lower height is good for your health. I saw a guy on the upper step of a step ladder that only had 3 legs touching the ground below, and he was so motivated to reach higher and more sideways to use the running chain saw in his 2 hands to cut an out-of-reach branch overhead........Well, I stepped up and grabbed the moving ladder to stabilize it and tried to "talk him down". Enjoy your future mulberry muffins.

I would prune it to a central leader but wait until late winter-early spring to do as you suggest, although I don't think a mulberry is as likely to split as trees with heavier fruit.

Mulberries tend to grow huge diametered branches that destroy the central leader shape by destroying the dominance of the leader while still quite small, but such a shape is more appealing to me and more efficient of space.

I would not only cut that branch but would use a piece of electric conduit to tightly tape the tree to with electric tape to make it grow straight. Then for the next few years I'd remove all branches that are more than half the diameter of the trunk at the point the branch is attached to that trunk.

I agree about using wood chips or anything to keep away the grass-weed competition. The tree is small for a 4 year old mulberry.

OK. Thank you. I'll take that under advisement.

i would also cut the tops and mulch heavily.

I tend to prune a bit different than other people i think.
i would probably keep that right branch, and where it forks out to 3 branches,
i would cut a foot or so above that.
this way it stays short,
but the forking provides more branches for more fruit to grow on.
mulberry trees grow fast, and can take pretty harsh pruning without damage.
(i still wouldnt cut more than 30% material though)

I would cut that left-most branch also about 2ft above the fork. and maybe even tie a weight to it, or a rope to bend it down to the ground, so when it gets taller, the branch is lower and easy to reach.
you can lookup - Espalier" to get some ideas.

Here is a link that might be useful: https://www.google.com/search?q=Espalier&tbm=isch