  We consider Coifman--Fefferman inequalities for rough homogeneous singular
integrals $T_\Omega$ and $C_p$ weights. It was recently shown by
Li-P\'erez-Rivera-R\'ios-Roncal that $$
  \|T_\Omega \|_{L^p(w)} \le C_{p,T,w} \|Mf\|_{L^p(w)} $$ for every $0< p <
\infty$ and every $w \in A_\infty$. Our first goal is to generalize this result
for every $w \in C_q$ where $q > \max\{1,p\}$ without using extrapolation
theory. Although the bounds we prove are new even in a qualitative sense, we
also give the quantitative bound with respect to the $C_q$ characteristic. Our
techniques rely on recent advances in sparse domination theory and we actually
prove most of our estimates for sparse forms.
  Our second goal is to continue the structural analysis of $C_p$ classes. We
consider some weak self-improving properties of $C_p$ weights and weak and
dyadic $C_p$ classes. We also revisit and generalize a counterexample by
Kahanp\"a\"a and Mejlbro who showed that $C_p \setminus \bigcup_{q > p} C_q
\neq \emptyset$. We combine their construction with techniques of Lerner to
define an explicit weight class $\widetilde{C}_p$ such that $\bigcup_{q > p}
C_q \subsetneq \widetilde{C}_p \subsetneq C_p$ and every $w \in
\widetilde{C}_p$ satisfies Muckenhoupt's conjecture. In particular, we give a
different, self-contained proof for the fact that the $C_{p+\varepsilon}$
condition is not necessary for the Coifman--Fefferman inequality and our ideas
allow us to consider also dimensions higher than $1$.
