CAUTION
If the multimeter lead to the machine ground happens to slip off, the meter will immediately jump to about 20 kV until something arcs to ground. If that happens to be the keyboard underneath the meter, it will die. If that happens to be you, you will be recommending your better points to your maker.
If you must use a multimeter, put it inside the machine with the leads firmly attached and stay away from it.
With that out of the way …
The meter appears to be set for AC milliamps, which is incorrect for measuring the tube current. Set it for DC milliamps with a fixed range over about 30 mA.
However, an Old School analog meter is better than a digital meter, because the tube current has a truly bizarre waveform that the analog meter basically filters out. Fancy digital meters show essentially random values, particularly for power settings under about 30% of the tube’s maximum rating.
Because that is the high voltage wire, do not use DIY insulation. The power supply anode connection must be firmly attached to one end of a high voltage connector and the tube anode wire should be firmly attached to the end side. If there are any intermediate connections, replace them with a single run of HV-rated wire from the tube to the HV supply connector.
If the anode wire is arcing to the machine frame, the arc will have charred the insulation and you must replace that part of the wiring with a complete run of new HV wire. DIY insulation repairs will not work with high voltage wiring.
The cathode wire is (generally, see below *) at a low voltage unless it becomes disconnected from ground, at which point high-voltage hilarity ensues. The cathode wire in my machine had a loosely twisted-and-taped joint that I replaced with a crimped connector.
Some previous discussions may be helpful:
(*) Very high-power laser tubes may have both a positive HV connection to the anode and a negative HV connection to the cathode. If that is the case for your machine, the cathode wire will have chunky high-voltage insulation just like the anode and that multimeter’s wiring can kill you stone cold dead.
Which arose in a 500 W (!) laser: