Initially Sweep, a dog, snubbed Sooty? This could be so (4)

I haven't cracked this one yet — but the brilliant team at Times for the Times have:

The first letter of Sooty, plus MUTT cut short (snubbed). The reference is to to Harry Corbett’s glove puppets, Sweep the dog and Sooty, a sort of bear that did magic tricks. Innocent days of TV.