Gravitational waves (GWs) provide a powerful, theory-independent probe of the dynamical structure of spacetime and the cosmological background. We investigate the propagation of linearized tensor modes in k-essence cosmology, where the dark sector is described by a non-canonical scalar field. Working in the high-frequency (short-wavelength) limit on a Friedmann-Lema^{}tre-Robertson-Walker background and restricting to the transverse-traceless sector, we obtain a modified evolution equation in which the GW speed remains strictly luminal (compatible with GW170817), while the background field $\bar{\varphi }$ generates a time-dependent effective mass term $m_{\rm eff}$. This background-induced mass alters the dispersion relation without introducing extra propagating degrees of freedom, producing an accumulated, frequency-dependent phase shift over cosmological distances. We show that $m_{\rm eff}$ is uniquely determined by background cosmological parameters and can be expressed as a redshift-dependent function $m_{\rm eff}(z)$, providing a direct connection between GW observables and scalar-field dynamics. The GW luminosity distance stays identical to the electromagnetic one, preserving standard-siren consistency. We confront the model with a joint Bayesian analysis combining cosmic chronometers, BAO, Pantheon+SH0ES, and standard-siren measurements from GWTC-2.1/3/4, and assess performance using ${\chi }^2$, $\chi^2_{\rm red}$, AIC, BIC, and DIC. The reconstruction is consistent with current data, reproduces the late-time expansion history, and provides data-driven constraints on the redshift evolution of $m_{\rm eff}(z)$.