In this paper, we show that many snarks have a shortest cycle cover of length 4/3 m + c for a constant c, where m is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of length 4/3 m + o(m). In particular, we prove that graphs with perfect matching index at most 4 have cycle covers of length 4/3 m and satisfy the (1,2)-covering conjecture of Zhang, and that graphs with large circumference have cycle covers of length close to 4/3 m. We also prove some results for graphs with low oddness and discuss the connection with Jaeger’s Petersen colouring conjecture.