DFA for words with $a$s not a multiple of 3 and at least one $c$
The product construction is a way of taking two DFAs for languages $L_1,L_2$, and constructing a new DFA for the language $L_1 \cup L_2$ or $L_1 \cap L_2$. In your case, $L_1$ consists of all words in which the number of $a$s is not a multiple of 3, and $L_2$ consists of all words containing at least one $c$. You have already constructed a DFA for $L_1$. Construct one for $L_2$, and use the product construction to construct one for $L_1 \cap L_2$.