  Bermond, Jackson and Jaeger [{\em J. Combin. Theory Ser. B} 35 (1983):
297-308] proved that every bridgeless ordinary graph $G$ has a circuit
$4$-cover and Fan [{\em J. Combin. Theory Ser. B} 54 (1992): 113-122] showed
that $G$ has a circuit $6$-cover which together implies that $G$ has a circuit
$k$-cover for every even integer $k\ge 4$. The only left case when $k = 2$ is
the well-know circuit double cover conjecture. For signed circuit $k$-cover of
signed graphs, it is known that for every integer $k\leq 5$, there are
infinitely many coverable signed graphs without signed circuit $k$-cover and
there are signed eulerian graphs that admit nowhere-zero $2$-flow but don't
admit a signed circuit $1$-cover. Fan conjectured that every coverable signed
graph has a signed circuit $6$-cover. This conjecture was verified only for
signed eulerian graphs and for signed graphs whose bridgeless-blocks are
eulerian. In this paper, we prove that this conjecture holds for signed
$K_4$-minor-free graphs. The $6$-cover is best possible for signed
$K_4$-minor-free graphs.
