For Mumbai, who had lost their opening match against Highveld Lions, it was a stop-start performance after they were sent into bat with wickets falling at regular intervals before Pollard threatened to cut loose at the later stage.
Pollard hit three fours and two sixes from the 20 balls he faced so far before rain came to stop the Mumbai innings in the Group B match at the Newlands.
None of the Mumbai batsmen could score big individually and the highest partnership was the 36-run stand between opener Dwayne Smith (37) and Rohit Sharma (25) for the second wicket.
The players came back into the field twice only to leave it immediately due to rain. When the first rain break came after the completion of the 17th over, Mumbai were 145 for six and Pollard hit a six from one of the five balls that were bowled before another rain break to take his side to 156 for six.
For Yorkshire, spinners Adil Rashid and Azeem Rafiq has so far taken two wickets apiece while Joe Root grabbed a wicket.
Earlier, Smith gave a dramatic start to the Mumbai innings as he smashed off-spinner Root, who opened bowling for Yorkshire, for two consecutive sixes from the second and third balls of the match.
The other opener Sachin Tendulkar (7) looked in good touch as he played a lofted drive for a boundary in the second over bowled by Ryan Sidebottom. But the next over, he had to depart run-out after a mix-up with Smith.
Tendulkar darted down for a single after playing a mistimed shot towards mid-on but with Smith not responding, he had to run back to his end. Before Tendulkar could regain his crease, Yorkshire wicketkeeper Dan Hodgson had dislodged the bails after collecting a quick throw from Gary Ballance.
