  The present work begins by examining the early-Universe inflationary epoch of
a special K-essence model, which incorporates a linear coupling term between
the scalar field potential and the canonical Lagrangian. For the power law
potential, we both numerically and analytically prove that the inflationary
parameters such as the spectral index and tensor-to-scalar ratio are compatible
with the recent BICEP/Keck observations. Continuing this work, our analysis
based on comparing early-Universe observations with late-Universe measurements
indicates that the tension on the Hubble parameter $H_0$ and the growth of
structure parameter $S_8$ can be alleviated simultaneously. More precisely,
compared to the standard $\Lambda$CDM model, our model can reduce $H_0$ tension
to roughly $2.2 \sigma$ and the $S_8$ discrepancy diminishes to $0.82\sigma$.
