  A long-standing conjecture of Berge suggests that every bridgeless cubic
graph can be expressed as a union of at most five perfect matchings. This
conjecture trivially holds for $3$-edge-colourable cubic graphs, but remains
widely open for graphs that are not $3$-edge-colourable. The aim of this paper
is to verify the validity of Berge's conjecture for cubic graphs that are in a
certain sense close to $3$-edge-colourable graphs. We measure the closeness by
looking at the colouring defect, which is defined as the minimum number of
edges left uncovered by any collection of three perfect matchings. While
$3$-edge-colourable graphs have defect $0$, every bridgeless cubic graph with
no $3$-edge-colouring has defect at least $3$. In 2015, Steffen proved that the
Berge conjecture holds for cyclically $4$-edge-connected cubic graphs with
colouring defect $3$ or $4$. Our aim is to improve Steffen's result in two
ways. First, we show that all bridgeless cubic graphs with defect $3$ satisfy
Berge's conjecture irrespectively of their cyclic connectivity. Second, we
prove that if the graph in question is cyclically $4$-edge-connected, then four
perfect matchings suffice, unless the graph is the Petersen graph. The result
is best possible as there exists an infinite family of cubic graphs with cyclic
connectivity $3$ which have defect $3$ but cannot be covered with four perfect
matchings.
