<p>This paper investigates a delayed Duffing equation with a fractional damping term. Conditions for Hopf bifurcation are derived, and the harmonic balance method is utilized to approximate periodic solutions. Analytical formulas are obtained for the amplitudes and frequencies of coexisting periodic solutions, revealing that their number can range from zero to infinity, controlled by the fractional derivative order and the time delay. Furthermore, the width of the existence region narrows as the number of coexisting solutions increases, and for high-frequency periodic solutions, the amplitudes vary linearly with their frequencies. The analytical results are validated through numerical simulations.</p>