By using unbiased determinant quantum Monte Carlo simulations, we investigate the interplay between superconductivity and charge density wave order in the Holstein model on a honeycomb lattice with next-nearest-neighbor hopping $t^{\prime }$. We find that a finite negative $t^{\prime }$ enhances $s$-wave superconducting pairing susceptibility near the van Hove fillings in the weak electron-phonon coupling regime, while it suppresses superconductivity and promotes charge density wave order at intermediate electron-phonon coupling strengths. The effect of $t^{\prime }$ on a charge density wave is filling dependent: it suppresses the charge density wave at half filling but enhances it near the van Hove singularities. A Spectral analysis reveals the opening of a gap at low temperatures, highlighting the competitive relationship between superconducting and charge density wave orders mediated by electron-phonon coupling and tuned by $t^{\prime }$.

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• Charge density waves
• Electron-phonon coupling
• Superconductivity
• Honeycomb lattice
• Holstein model
• Hybrid Monte Carlo algorithm