Here are a couple of character theoretic observations which do not require CFSG.
In the situation of the question (where $C$ denotes the conjugacy class of $g$), we have $|C \cap Hx| = 1$ for every $x \in G$ if and only if $\langle {\rm Res}^{G}_{H}(\chi), 1 \rangle = 0$ whenever $\chi \in {\rm Irr(G)}$ is a non-trivial character with $\chi(g) \neq 0.$ More generally, $\sum_{ t \in T} |C \cap Ht|^{2} = [G:H] \sum_{\chi \in {\rm Irr}(G)} \frac{|\chi(g)|^{2}}{\chi(1)} \langle {\rm Res}^{G}_{H}(\chi,1 \rangle,$ where $T$ is a transversal to $H$ in $G.$

This formula is derived by considering the product (in the group algebra $\mathbb{Z}G$), of class sums ${\tilde C}{\tilde C^{-1}},$ where $C^{-1}$ denotes the class of $g^{-1}$ and we use ${\tilde C}$ to denote the class sum of the class of $g.$ The coefficient of $x \in G$ in this product is well-known to be $\frac{|G|}{|H|^{2}} \sum_{ \chi \in {\rm Irr}(G)} \frac{|\chi(g)|^{2} \chi(x^{-1})}{\chi(1)}.$ This is always non-negative, and if we sum these quantities over $x \in H,$ the claims follow easily ( in the first case, the trivial character already contributes $[G:H]$ to the RHS, and all other terms on the RHS are non-negative. In the second case, the conjugates of $g$ in $Ht$ contribute $|C \cap Ht|^{2}$ elements of $H$ to the given product of class sums, including multiplicities).

Note that we easily obtain $\sum_{t \in T}(|C \cap Ht|-1)^{2} \leq d(|G|-[G:H]),$ where $d$ is the maximum value over non-trivial irreducible characters $\chi$ with $\chi(g) \neq 0$ of $\frac{\langle {\rm Res}^{G}_{H}(\chi), 1 \rangle}{\chi(1)}$.

Later edit: In fact, this problem is quite closely related to an earlier one on MO about doubly transitive action on a conjugacy class: if $p$ is a prime, and $G$ is a (putative) doubly transitive permutation group whose point stabilizer $H$ has a non-trivial center (and with $F(G) =1$), then it can be shown that for an element $z \in Z(H)$ of prime order, there is one conjugate of $z$ in each coset of $H,$ ie the conjugates of $z$ form a transversal to $H.$

Also, I mention (without proof, but in case it is useful to anyone else), the following facts which may be proved using block theory: if $G$ is a finite group and $z \neq 1$ is an element of order a power of a prime $p$ whose conjugates form a transversal to $H = C_{G}(z),$ the following hold:

Whenever $y$ is a $p$-regular element of $H,$ we have ${\tilde C}_{zy} {\tilde H} = [H:C_{H}(y)]{\tilde G}$ (where, as before, for $S$ a subset of $G,$ we let ${\tilde S}$ denote the sum of the elements of $S$ in the group algebra $\mathbb{Z}G,$ and where $C_{u}$ denotes the conjugacy class of $u.$

Whenever $x$ is a $p$ -element (possibly the identity element)of $H,$ we have $|S_{p}^{G}(x)| = [G:H]|S_{p}^{H}(x)|$
and ${\tilde S}_{p}^{G}(x){\tilde H} = |S_{p}^{H}(x)|{\tilde G},$ where $S_{p}^{H}(x)$ denotes the $p$-section of $x$ in $G$ ( that is, the set of elements of $G$ whose $p$-part is conjugate to $x$).

By way of explanation, these last facts follow because the trivial character is the only constituent in the principal $p$-block of the character ${\rm Ind}_{H}^{G}(1),$ and central characters associated to irreducible characters outside the principal $p$-block annihilate $p$-section sums by Brauer's Second Main Theorem. For the first, we also have the more precise fact that the trivial character is the only constituent of ${\rm Ind}_{H}^{G}(1)$ in a $p$-block with a defect group containing $z,$ so all non-trivial constituents of ${\rm Ind}_{H}^{G}(1)$ vanish on the $p$-section of $z.$