When the click stops
Know who keeps the song moving if the reference disappears.
If the click stops during a song, keep playing a part you can hold while you listen and look for the leader. Avoid hunting through a device while the band is still depending on you. The next step depends on whether only your monitor failed or the whole playback system stopped.
Agree on a plan before the set
Ask who controls playback and what the band will do if it fails. A song with backing tracks may need a restart or another cue; a song played entirely live may continue. Agree how that decision will be signaled.
At rehearsal, stop the click during a familiar section and practice the response. Keep the groove simple and listen to the bass, rhythm instrument and vocal. Record the test if possible, then check whether the band stayed together through the ending.
Treat monitor loss separately
If your monitor goes silent while the rest of the band continues, watch the leader and listen to the sound available in the room. Do not assume everyone lost the click. Signal the fault when you can do so without missing a cue, and let the agreed plan guide the response.
Between songs, check the relevant connections, power and battery with the person responsible for that equipment. A phone metronome only helps if the band has agreed how to use it and it can be heard through the intended monitor path.
Keep a playable alternative
If a hi-hat stand becomes unusable, ride time can provide an alternative until a pause. If a drum or stand fails, adapt only if the remaining setup is safe and stable. Rehearse a plain version of the part so an equipment problem does not also become a new arrangement you have to invent under pressure.