The Waikato rider was 16th in the individual pursuit in 3:47.448, 18th in the scratch race and 12th in the elimination.
Sheath lies in 16th place with the hometown star Laura Trott (GBR) leading on 118 points from Kirsten Wild (NED) on 116.
Earlier Eddie Dawkins and Matt Archibald placed second in their heats of the keirin, with Dawkins winning his repechage to advance to the second round. However he was fourth in his second round race and was eliminated while Archibald missed out in the repechage.
Stephanie McKenzie qualified for the elimination rounds of the women's sprint, clocking 11.406 to be 20th in the qualifying led by Elis Ligtlee (NED) 10.833.
McKenzie lost her first round of elimination to Hong Kong's Wai Sze Lee.
In non-world cup race, Westley Gough and Pieter Bulling were fourth in the men's Madison.
Tomorrow Gate and Sheath complete the omnium competition while the men are in the individual sprint, with the women in the keirin.