A moving lemma for cycles with very ample modulus

We prove a moving lemma for higher Chow groups with modulus, in the sense of Binda-Kerz-Saito, of smooth projective schemes when the modulus is given by a very ample divisor. This provides one of the first cases of moving lemmas for cycles with modulus, not covered by the additive higher Chow groups. We apply this to prove a contravariant functoriality of higher Chow groups with modulus. We prove a localization sequence for cycles with modulus and use this to describe certain relative $K$-theory in terms of cycles with modulus.