Suppose $f_1(n)=n\log^*n$,$f_2(n)=n\log h$ that $h$ is number of vertices of convex hull.
Can we conclude that $$f_1+f_2=O(f_2)?$$ Edited:
Note that, $h$ is a function accroding to $n$ that $h\leq n$, and according to the input $n$ it can be less than $n$ such as constant.
$n\log h$ is running time of an ouput sensetive algorithm for finding convex hull.
$h$ is the number of vertices of what convex hull?
If $h$ is a constant, then $f_2 = \mathcal{O}(f_1)$ because $\log^*n\underset{n\rightarrow +\infty}{\longrightarrow}+\infty$.
If I try to interpret more than you wrote, and consider $h$ to be the number of vertices of the convex hull of a set of $n$ points, then you could still have $f_2 = \mathcal{O}(f_1)$, because $h$ could be equal to $3$, even with $n$ arbitrary big.
