  The study of pattern avoidance in linear permutations has been an active area
of research for almost half a century now, starting with the work of Knuth in
1973. More recently, the question of pattern avoidance in circular permutations
has gained significant attention. In 2002-03, Callan and Vella independently
characterized circular permutations avoiding a single permutation of size $4$.
Building on their results, Domagalski et al. studied circular pattern avoidance
for multiple patterns of size $4$. In this article, our main aim is to study
circular pattern avoidance of $[4,k]$-pairs, i.e., circular permutations
avoiding one pattern of size 4 and another of size $k$. We do this by using
well-studied combinatorial objects to represent circular permutations avoiding
a single pattern of size $4$. In particular, we obtain upper bounds for the
number of Wilf equivalence classes of $[4,k]$-pairs. Moreover, we prove that
the obtained bound is tight when the pattern of size $4$ in consideration is
$[1342]$. Using ideas from our general results, we also obtain a complete
characterization of the avoidance classes for $[4,5]$-pairs.
