It's not a click noise. It's more like a one loud grunt.
ser# XXXXX?88, model # XXXXX (not real legible)
It appears that the only time it does this is when the refrig cycles and it shuts off. It sounds like the compressor to me. It is still running okay.
OK so this is going to be your compressor. It means the piston inside has broken free so when it shuts of it bangs against the side of the compressor shell. I've seen refrigerators last for years and years with this problem, but eventually the piston will get jammed one day and suddenly stop cool. The only fix is to replace the compressor which is about $550 normally.
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