What do you have the power supply's current knob set to when experiencing this fault and conducting these tests? If you're artificially limiting the power supply's ability to deliver current to the load, then it has no choice but to reduce its output voltage to obey your command.
As PKTKS pointed out, you have a potential short circuit being created with the multimeter's incorrect setup. This may have led you to turn down the current limit on the power supply.
First, fix the multimeter by ensuring that the leads are inserted into the correct sockets for the type of measurement you wish to make (e.g., Com and V for voltage) and set the mode to match. Then, set the power supply to maximum current and the voltage to the desired level (e.g., 5 volts). Measure.
Next, set the power supply's voltage to what's needed by the load that you want to test. Keep the current setting at maximum. Connect the load and see if it works properly.