= Solution
Declare an edge of the <planar dual graph> open exactly when its crossed primal edge is closed. This <dual bond percolation> has parameter $q=1-p<1/2=p_c$, using the <Harris-Kesten theorem>. We use the standard <exponential tail of subcritical cluster size>: for some $K,a>0$, uniformly in the dual vertex $v$,
$$
\mathbb P_q(|C^*(v)|\geq k)\leq Ke^{-ak},\qquad k\geq1.
$$
This is decay of the number of vertices in the <percolation cluster>, not merely its radius.
On $\{|C|=n\}$, the geometric fact allowed in the paper supplies a simple dual <cycle in a graph> surrounding $C$, with length $\ell\geq\alpha\sqrt n$. Every crossed edge is in the <edge boundary> of $C$ and is therefore closed, so the dual cycle is open. Its $\ell$ distinct vertices lie in one dual <percolation cluster> of size at least $\ell$.
A dual <cycle in a graph> of length $\ell$ surrounding the origin has all its vertices within sup-norm distance $\ell+1$ of the origin: its coordinate spans are at most $\ell$, and the origin lies between each pair of extreme coordinates. There are at most $K_0(\ell+1)^2$ possible dual vertices there. The <union bound> over lengths and possible vertices yields
$$
\mathbb P_p(|C|=n)\leq KK_0\sum_{\ell\geq\lceil\alpha\sqrt n\rceil}(\ell+1)^2e^{-a\ell}\leq K_1e^{-b\sqrt n}
$$
for some $K_1,b>0$, since a polynomial factor can be absorbed into a slower <exponential decay>.
To remove the constant prefactor for all $n\geq1$, first choose $n_0$ so that the bound is at most $e^{-(b/2)\sqrt n}$ for $n\geq n_0$. For the remaining finitely many $n$, use $\mathbb P_p(|C|=n)\leq1-\theta(p)<1$ and choose
$$
\boxed{c_2=\min\left\{\frac b2,\frac{-\log(1-\theta(p))}{\sqrt{n_0}}\right\}>0,\qquad
\mathbb P_p(|C|=n)\leq e^{-c_2\sqrt n}.}
$$
For $1\leq n<n_0$, the chosen $c_2$ ensures $e^{-c_2\sqrt n}\geq1-\theta(p)$.
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