for testing: remove the anti back fire solenoid. (yes, fuel will drain out, have a small container handy to catch fuel). With fingers or the back of plastic handle screw driver, depress the pin into the solenoid. (engine OFF). Do this a few times. If it will not budge spray some wd-40. Try it again. Now, with anti back fire solenoid removed, key to start and you should be able to see the pin retract as it should. It's hard to listen to it click because you might confuse the click with the starter solenoid, which also can give a slight click sound too...so just watch the pin and see if it will retract. if not, try excercising the pin over and over with wd-40...try again...if it still will not budge, then you have some option:
1. meter the connector while you start and see if there is 12 volts. If there is no voltage, then you have a power supply problem. Go all the way back to the disconnect...usually there is a large disconnect somewhere between the anti backfire solenoid connector and the ignition switch. check the power there...if there IS power there, then you probably either have a bad connector or open wire between the connector and the anti backfire solenoid. often this is the wires getting crushed from factory zip ties somewhere along the route. Look at this zip tied locations. Sometimes the wires get borked and insulation loss across the engine or engine shroud, causing a short. Basically this option is looking for why you are not getting power to the anti back fire solenoid connector. If you are not getting power to the disconnect on ENGINE START, the continue tracing back from that connector to the ignition switch...looking for wire damage or a corroded or grounded spade connector on the ignition switch. (as preventative maintenance, I wd-40 ignition switches at least once a year..internally, they can corrode as well as the backside where the terminals can get junked up from debris, causing shorts between wiring connectors.
2. if you do have power to the anti back fire solenoid connector (at engine start), and there are no issues traced all the way back to ignition switch, then it's more than likely a faulty anti back fire solenoid device...either it's so gummed up that the solenoid failed, or as is the case usually these are cheaply made prone to fail devices that die over time.. You can if you want to, just delete the anti back fire solenoid from the engine and get a fine threaded bolt to hold the bowl of the carb with a simple bolt gasket to prevent leaks. If you are concerned about back fires, idle down before shutting engine off. I've done this many times and there is no downside to just deleting the device. If the engine pops in shutdown, I'm not losing sleep over it. But it rarely happens. anti back fire solenoids are expensive, but you can sometimes find alternate part numbers by cross referencing the carb..(versus trying to do it with the engine or mower model). So if you find the carb model number, look for the anti back fire solenoid that fits it and matches the wiring connector you have. Sometimes you find that there are many different part numbers for anti back fire solenoids that work for a carb. if you follow the mower parts list or engine parts list from the manufacturer they do not generally give you alternate parts that will actually work..and that are cheaper. I'm sure we can assume why they would not give this info out. :-(