that click followed by a hum can only be one thing
pull the unit out and feel the compressor, if its really hot you have a relay and overload issue, and it struggles to start.
with any luck, it is the external relay and overload found mounted to the comresssor should be a lil white block
if not it will be two differant parts
a lil white one and a brown one
let these cool a bit and check both for continutity
ive included a link for the parts
they are 10 and 11 on the schematic
http://www.appliancepartspros.com/partsearch/model.aspx?model_id=5174380&diagram_id=1358665#d1358665