**It is consistent that the answer is no**:

Let $V=L[U],$ where $U$ is a normal measure on a measurable cardinal $\kappa.$ First note that we can apply Prikry forcing over $V$ to change the cofinality of $\kappa$ to $\omega.$ 

Now we show that there is no forcing of size $\kappa$ changing the cofinality of $\kappa.$ Suppose not. Let $P$ be such a forcing notion and let $G$ be $P-$generic over $V$. By Dodd-Jensen covering theorem for $L[U],$ there exists an $\omega-$sequence $C\in V[G]$ cofinal in $\kappa$ which is a Prikry sequence
for the classical Prikry forcing $P_U$ in $V$ and $V[G]$ is covered by $V[C]$. Then $V[C]\subset V[G],$ and we have $P_U$ is a projection of $P$, so $P$ has size $>\kappa.$ 


**Remark.** In fact the above proof shows that if there is such a forcing notion, then $0^\dagger$ exists.