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That clicking noise is the compressor overloading and cutting out. If you listen close next to the compressor on the back, you'll see that the clicking is coming from the side of the compressor. This is not something you can just reset. When the compressor cools down (like unplugging the unit for awhile) it will work normally until the compressor warms up again. Either the overload relay on the side of the compressor is getting weak and needs to be replaced or the compressor is just starting to overheat and needs to be replaced. You can't replace the compressor yourself so I would just replace the overload relay since you can't test to determine if it's weak or not. The part number you'd need to order for your model is EBG44336201. It's the small plastic box device attached to the side of the compressor where the wires plug in. If this doesn't solve the problem, you'd need a technician to replace the compressor which is about $650 I'm sorry to say.
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Most likely the parts would not be available in a local store, but you can order them at Searspartsdirect.com if needed. If the problem is just the overload relay I would definitely just fix it because the cost is maybe $50 for the part. But if you need a new compressor, I might think about just replacing the appliance rather than repairing it.
Hello again. I'm just following up because I haven't heard back from you and I see this question is still open. Is there anything else I can help you with?