A longstanding conjecture of Alon and Tarsi, and independly Jaeger (1985), suggests that the edges of every bridgeless graph can be covered with cycles of total length at most 7/5 • m, where m is the number of edges. We study the relationship between cycle covers and structural properties of cubic graphs, focusing on their colouring defect. This invariant, introduced by Steffen in 2015, is defined as the minimum number of edges left uncovered by any set of three perfect matchings of a cubic graph. We show that every bridgeless cubic graph with colouring defect not exceeding 3 admits a cycle cover of length at most 4/3 • m + 1, just one step above the universal lower bound of 4/3 • m for all cubic graphs. We also prove that, regardless of defect, the same bound holds for bridgeless cubic graphs that have an edge whose endvertices removed yield a 3-edge-colourable graph and the edge lies on a 5-cycle. Motivated by our investigations, we introduce a new invariant for cubic graphs, their covering excess, to measure the deviation of the length of a shortest cycle cover from the mentioned lower bound. Finally, we show that every bridgeless cubic graph with covering excess at most 1 admits a cycle double cover.
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