Laura Coombs and Christie Murray were also on the mark for Neil Redfearn's new-look side.
Liverpool FC Women took the lead on 28 minutes when Murray was brought down and converted the resulting penalty.
Courtney Sweetman-Kirk was then unfortunate to see her close range header brilliantly saved and pushed onto the bar.
It was 2-0 on 58 minutes when Clarke found the net with an unstoppable header following a great cross from Murray.
Murray played a part in the third goal on 65 minutes when a sweet one-two saw Coombs slot home the final goal.
All eyes now turn to Liverpool FC Women's opening game of 2018-19, when they host Manchester United in the group stage of the Continental Cup on August 19 at Prenton Park (2pm BST).