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In the study of algebraic curves and their moduli spaces, it is important todetermine the $a$-numbers of curves over a field of positive characteristic. Itis known that non-hyperelliptic curves of genus $3$ are classified by thestructures of their automorphism groups as finite groups. In this paper, wedetermine the $a$-numbers of non-hyperelliptic curves of genus $3$ with cyclicautomorphism group of order $6$ or $9$. Moreover, we also find the exact numberof the isomorphism classes of such curves attaining the possible maximal$a$-number.

The $a$-numbers of non-hyperelliptic curves of genus three with large cyclic automorphism group