  We investigate the low moments $\mathbb{E}[|A_N|^{2q}], 0<q\leq 1$ of
{secular coefficients} $A_N$ of the {critical non-Gaussian holomorphic
multiplicative chaos}, i.e. coefficients of $z^N$ in the power series expansion
of $\exp(\sum_{k=1}^\infty X_kz^k/\sqrt{k})$, where $\{X_k\}_{k\geq 1}$ are
i.i.d. rotationally invariant unit variance complex random variables. Inspired
by Harper's remarkable result on random multiplicative functions, Soundararajan
and Zaman recently showed that if each $X_k$ is standard complex Gaussian,
$A_N$ features better-than-square-root cancellation: $\mathbb{E}[|A_N|^2]=1$
and $\mathbb{E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$ for fixed $q\in(0,1)$ as
$N\to\infty$. We show that this asymptotic holds universally if
$\mathbb{E}[e^{\gamma|X_k|}]<\infty$ for some $\gamma>2q$. As a consequence, we
establish the universality for the tightness of the normalized secular
coefficients $A_N(\log(1+N))^{1/4}$, generalizing a result of Najnudel,
Paquette, and Simm. Another corollary is the almost sure regularity of some
critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces.
Moreover, we characterize the asymptotic of $\mathbb{E}[|A_N|^{2q}]$ for
$|X_k|$ following a stretched exponential distribution with an arbitrary scale
parameter, which exhibits a completely different behavior and underlying
mechanism from the Gaussian universality regime. As a result, we unveil a
double-layer phase transition around the critical case of exponential tails.
Our proofs combine Harper's robust approach with a careful analysis of the
(possibly random) leading terms in the monomial decomposition of $A_N$.
