data structure admitting top items with prefix

Question

A trie can answer a query for all items with a certain prefix in $O(m + \log n)$ time, where $m$ is the number of matches and $n$ is the number of items. A trie also supports $O(\log n)$ insertion and deletion and takes $O(n)$ space.

Now suppose each item has a score. I am looking for a data structure that can answer a related query: give me the top $m$ items by score with a certain prefix. Ideally search, insertion, and deletion would have the same time as a trie, up to a constant factor, and space would be $O(n)$ as well. To this end we may assume items are short (of length $O(\log n)$).

This problem can be solved with $O(\log^2 n)$ insertion and deletion and $O(n \log n)$ space by forming a trie where every node contains a sorted set of items with that prefix sorted by score. But this approach uses excess time and space compared to a trie. Can we do better?