Diagnosis
Smart fear. The Branch desk's built-in cable management tray is a plastic channel that runs across the back underside of the desktop. Clamping a monitor arm directly over it will crack it.
How to Fix It
You need to choose your clamp location carefully. Avoid the back 4-6 inches of the desk entirely. The safest zone is usually near one of the legs, where the metal frame of the desk is directly underneath the desktop, providing solid support. Before you clamp, feel underneath to map out where the plastic channel is. If your desktop is the Branch laminate type, the core is still strong particleboard, so a clamp in a clear area is fine.
If That Didn't Work
If your ideal viewing position forces you to clamp near the back, consider a different mounting solution. A monitor arm that uses a grommet mount (a hole through the desk) is the most secure. You can drill a hole (carefully!) in a clear spot, which is safer than risking the clamp. Or, use a stand that sits on the desk surface instead of clamping.
Heads Up
When you tighten the clamp, don't go full Hulk. Tighten it just enough that the monitor doesn't wobble. Overtightening can dent or delaminate the desktop even in a safe spot. Use the included rubber pads to protect the finish.