What is partial expansion in Monte Carlo Tree Search?
Partial expansion is a pruning/reduction strategy. Instead of requiring all children to be explored in selection you could set a threshold like done in "Move Count Pruning" (MCP) based on only expanding the first 10 children only. Usually you call ucb_sample if all children are explored depending on your weights. 

Another example could be based on the rollout. Even rollouts need full expansion and odd rollouts need only MCP. This way the tree gets explored in a different way.

Benefits usually come in reducing time and favouring early moves. Early moves get more time and those nodes get better statistics. With backpropogation grandchildren help update their parents because statistics exist.