The connection between the desk and the T-bar stand is just four thin metal standoffs that are formed from a stiffener bracket welded to the back of the desk. The thin metal over time with use, flexes and breaks away due to metal fatigue.
This repair simply replaces the standoffs using commonly available hardware.
4-1/4” threaded rod coupling about 7/8 " long
4-1/4" x 3/8" truss head screws (wide flat head)
4-1/4" flat washers
The photo shows both the new parts and the old parts together.
The tools include
socket wrench to fit coupling (standoff)
vice or clamp
Then using the pliers break away any remaing standoffs.
then hammer flat any remnants of the standoffs.
Then be sure to square up the desk and stand.
Using a T-bar bracket as a guide centerpunch the standoff locations.
with the 1/4" bit drill the holes for the truss screws
de-burr those holes
As I have a number of these to do I created a plywood template to mark the standoff location.
Add a washer, then screw on the standoff.
Tighten all of these very tight.
At this point i use my template to make sure that the top of the standoffs are properly located
You may need some minor adjustments
Put the bracket in a vice or somehow secure it so that it cannot spin as you drill this hole..
Once that is done we can reassemble the desk with the T-stand