Harry didn't 'throw his weight' onto me. He wasn't over-assertive towards me. He didn't apply pressure to make me leave early, or to make me choose him to inherit my job. He waited patiently for me to leave and waited patiently to see if he would take over the company after me.
This is what it means, within this context. But really, it's a poor choice of wording. It could be a lot clearer. It's intentionally worded in an abstract manner, which would make it difficult for any non-native English speaker to understand, or even for any native speaker to understand if they don't have a good grip on the context by reading plenty of surrounding text.