The major sellers of Mastech meters are Amazon, eBay, AliExpress, Konga, Joom, Indiamart and wish.com. That is a huge red flag for cheap counterfeit crud, right off the boat.
If that fact did not reflect the price you paid, that's because crud sellers have discovered people are more likely to buy it if it's up-priced comparable to quality units. Also Mastech is a real brand, they just don't sell test equipment. This "brand predation" is very typical of "those guys".
I bought a modest quality Klein unit last year for around $50, so I don't see any reason to pay that much for junk.
The lamp didn't produce a reading because its current draw is negligible - perhaps 0.1 amp assuming you are using modern bulbs.
If you cut up any line cords to wrap the meter around 1 wire, you need to replace those line cords, or shorten them and replace with a UL-listed retrofit plug.
The normal way of testing cord-and-plug connected equipment without slicing up the cord is to use a line splitter as Doxylover discussed. Some just spread the wires for you, for a 1x reading. Others wrap the wires 10 times around the hole, giving you a 10x reading -- helpful for things like that lamp.
10x makes sense. Most clamp ammeters range to at least 200A, and 20A is the max for anything that plugs into this type of socket.
Realistically I think you got much too large a clamp meter for residential-tier activities. If quality were equal, a larger "max limit" means poorer resolution on small measurements.
