The differential affine velocity estimator for vector magnetograms (DAVE4VM) is widely used to infer the photospheric magnetic energy and helicity flux in solar active regions (ARs). Here, we incorporate the observed Doppler velocity as a new constraint. We test the performance of the new algorithm, DAVE4VMwDV (DAVE4VM with Doppler Velocities), on synthetic data from the anelastic magnetohydrodynamic (ANMHD) simulations. We find that it recovers roughly 90% of the total helicity flux and 95% of the total Poynting flux from the ANMHD ground truth. We apply DAVE4VMwDV to observations of AR 11158, a well-studied AR located close to the disk center. Owing to the new constraint, the evolution of Poynting and helicity flux differ significantly when compared to DAVE4VM. The discrepancy is also significant when we compare a few commonly used algorithms. Interestingly, we find that our new helicity estimate is close to another algorithm, Doppler-FLCT-Ideal (DFI), which also utilize Doppler data explicitly. We speculate that the Doppler velocity measurement is important for the magnetic energy and helicity flux estimate; a careful treatment of various systematic uncertainties is crucial.