Context. Recent works highlight the importance of stellar X-rays on the evolution of the circumstellar disks of young stellar objects, especially for disk photoevaporation. Aims. A signature of this process may be seen in the so far tentatively observed dependence of stellar accretion rates on X-ray luminosities. According to models of X-ray driven photoevaporation, stars with higher X-ray luminosities should show lower accretion rates, on average, in a sample with similar masses and ages. Methods. To this aim, we have analyzed X-ray properties of young stars in the Orion Nebula Cluster determined with Chandra during the COUP observation as well as accretion data obtained from the photometric catalog of the HST Treasury Program. With these data, we have performed a statistical analysis of the relation between X-ray activity and accretion rates using partial linear regression analysis.Results. The initial anticorrelation found with a sample of 332 young stars is considerably weaker compared to previous studies. However, excluding flaring activity or limiting the X-ray luminosity to the soft band (0.5 − 2.0 keV) leads to a stronger anticorrelation, which is statistically more significant. Furthermore, we have found a weak positive correlation between the higher component of the plasma temperature gained in the X-ray spectral fitting and the accretion rates, indicating that the hardness of the X-ray spectra may influence the accretion process. Conclusions. There is evidence for a weak anticorrelation, as predicted by theoretical models, suggesting that X-ray photoevaporation modulates the accretion rate through the inner disk at late stages of disk evolution, leading to a phase of photoevaporation-starved accretion.