Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define $q_i=\frac{a_i}{C+\sum_{i=1}^na_i}$ and $q_{n+1}=\frac{C}{C+\sum_{i=1}^na_i}$. Does Shannon entropy of $q$ dominate entropy of $p$?

Take $C=a_n+\log_2^ka_n$ for any fixed $k$. Now does Shannon entropy of $q$ dominate entropy of $p$ after certain $n$?

There are two cases to consider. Case $(1)$ $a_{i+1}=a_i+O(\log^ka_i)$ Case $(2)$ $a_{i+1}=a_i+O(a_i)$.

When can one expect Shannon entropy to be dominant in sequences of these types? Clearly in this post https://mathoverflow.net/questions/191391/entropy-dominance, one criteria  for negative result is given.