Assuming fair coin flips, I know how to compute the expected number of coin flips to see HTH sequence by writing out the linear system of equations from the state transition diagram below.
https://i.stack.imgur.com/N3YGa.png
Define $E(X)$ as the expected number of steps from state X to state HTH. E(0) means expected number of steps to get from state 0 to state HTH and E(H) means expected number of steps to get from state H to state HTH, etc. I'd like out the equations with the transition probability and expected value of the neighbor state plus one for the immediate step. The full linear system of equations is as follows.
\begin{align} E(0) &= \frac{1}{2} (E(H) + 1) + \frac{1}{2} (E(0) + 1)\\ E(H) &= \frac{1}{2} (E(H) + 1) + \frac{1}{2} (E(HT) + 1)\\ E(HT) &= \frac{1}{2} (E(HTH) + 1) + \frac{1}{2} (E(0) + 1)\\ E(HTH) &= 0 \end{align}
We can solve for $E(0)$ since we have 4 equations and 4 unknowns. It turns out $E(0)=10$
The question is how do we compute the variance $V(0)$. I understand that we can build the transition matrix and use the one line matrix formula from Wikipedia. However, I'm interested in intuitive system of equations approach (without any of Markov chain jargons like fundamental matrix, ergodic, transient, etc) just like the one I wrote above to compute $E(0)$. My hunch is we need systems of equations defined with $E(X^2)$ variables but it's not clear to me how to approach that.
In order to obtain the variance we have to count more carefully. Again we have to consider the states $0$, $H$, $HT$, and $HTH$. Denote the probability that in state $H$ the game is over in exactly $k$ additional moves by $p_H(k)$, and define $p_0(k)$, $p_{HT}(k)$, $p_{HTH}(k)$ similarly. Then we have to solve the system $$\eqalign{ p_0(k)&={1\over2}p_H(k-1)+{1\over2}p_0(k-1) \cr p_H(k)&={1\over2}p_{HT}(k-1)+{1\over2}p_H(k-1) \cr p_{HT}(k)&={1\over2}p_{HTH}(k-1)+{1\over2}p_0(k-1)\cr p_{HTH}(k)&=\delta_{0k}\cr}$$ along the lines of the "Master Theorem" and will then have an explicit expression for $k\mapsto p_0(k)$. This will allow to compute what you call $V(0)$.