Relay causes USB sound card to freeze
This may not be the answer, but I see no way of embedding a schematic in a comment...

Look for loop area in the mains switching circuit, and minimise it. Here's what I mean. Try to turn this circuit...







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into this circuit







simulate this circuit
The tests you have done so far suggest the interference may be coming from the switched mains circuit, but only if the relay (not the original switch) is used. One possible source is untidy layout, creating a large loop area, which increases the radiated energy of a switched current. 

The second current routes live and neutral together where possible (indeed, twisting them can reduce the area further) and reduces radiated interference.

It is also possible that the relay contacts bounce more than the switch, so connecting an X2 rated (safety) capacitor of 0.1 uF across L and N after the relay may also suppress interference.