We propose a new class of $f(R)$ theory where its Weyl gauge symmetry is
broken in the primordial era of the universe. We prove that, even though the
theory is transformed into the Einstein-Hilbert action with a non-minimally
coupled scalar field at the non-perturbative level, there exists an additional
non-minimal coupling at the perturbational level. As an important example, we
study its effect on Staronbinsky inflation. We show that the amplitude of the
primordial gravitational waves also affects scalar perturbation due to the
presence of the non-minimal coupling, although its effect on cosmic microwave
background(CMB) anisotropy is negligible in practice. Consequently, CMB
observables may have distinct values depending only on the mass of the
perturbed Weyl field. Moreover, we discuss the possibility of resolving Hubble
tension with this example, including an analysis of the integrated-Sachs-Wolfe
effect.