What is causing a voltage drop on the 5v power rail on this bench setup?
Your problem: the series resistance of wires and contact resistances.

It looks to me that 1.24 A is flowing. You're using crock-clips, thin wires with "fragile" breadboard type connectors. That can easily cause a significant voltage drop at that current.

Confirm that by measuring at the supply output directly (you should get 5 V), then at the banana-crock clips (you will still get about 5 V), then at the crock-clip-thin wire (I guess you will get 4 V), then at the gold contacts on the card. Each measurement will give a lower voltage.

I suggest to solder some wire to the +5 V and ground rails, for example in parallel with one of the many decoupling capacitors on the card. Then make a solid connection from those wires to the supply. Maybe the supply has screw terminals, then use those. Prevent clips, connectors and contacts as much as possible.