  The colouring defect of a cubic graph, introduced by Steffen in 2015, is the
minimum number of edges that are left uncovered by any set of three perfect
matchings. Since a cubic graph has defect $0$ if and only if it is
$3$-edge-colourable, this invariant can measure how much a cubic graph differs
from a $3$-edge-colourable graph. Our aim is to examine the relationship of
colouring defect to oddness, an extensively studied measure of uncolourability
of cubic graphs, defined as the smallest number of odd circuits in a
$2$-factor. We show that there exist cyclically $5$-edge-connected snarks
(cubic graphs with no $3$-edge-colouring) of oddness $2$ and arbitrarily large
colouring defect. This result is achieved by means of a construction of
cyclically $5$-edge-connected snarks with oddness $2$ and arbitrarily large
girth. The fact that our graphs are cyclically $5$-edge-connected significantly
strengthens a similar result of Jin and Steffen (2017), which only guarantees
graphs with cyclic connectivity at most $3$. At the same time, our result
improves Kochol's original construction of snarks with large girth (1996) in
that it provides infinitely many nontrivial snarks of any prescribed girth
$g\ge 5$, not just girth at least~$g$.
