The graph \(C_n(d; i, j; P_k)\) denotes a cycle \(C_n\) with path \(P_k\) joining two nonconsecutive vertices \(x_i\) and \(x_j\) of the cycle, where \(d\) is the distance between \(x_i\) and \(x_j\) on \(C_n\). In this paper, we obtain that the graph \(C_n(d; i, j; P_k)\) is strongly \(c\)-harmonious when \(k = 2, 3\) and integer \(n \geq 6\).