And, if you do, it's usually written as 'softly, softly catchee monkey' - as TCL suggests - as if someone, perhaps Chinese, is speaking pidgin English. It first appeared in print in a book of quotations published in 1907. It is just a proverbial phrase recommending caution and the gentle approach as the best way to achieve an objective.
This is from the TV programme quiz because it has been asked and answered over and over again in Quizzes and Puzzles.The general consensus was that the answer is " Softly Softly " a spin off from Z Cars.
But ...whoever set this quiz should know that softly softly catches monkey...not mousey as TCL and QM state.
According to this BBC website http://www.bbc.co.uk/dna/h2g2/A10832474
softly softly catchee monkey was a phrase picked up by Baden Powell during a military campaign he was involved in in Ghana.