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.

Dan's answer is a good one. It's difficult to know what the author's exact meaning is based on these few sentences. I'm going to suggest a possible alternative meaning. When someone is over us, they are higher in authority and prestige -- such as a boss, a parent, or a more powerful person. It's possible that Harry didn't use his superior position to pressure the speaker. Another term we use for this is 'to put oneself above someone'. I think the addition of the word 'on' make's Dan's answer more likely. 'Putting oneself over ON someone' implies pressure rather than just authority alone. Another meaning of this phrase is to deceive. In this case, what is being 'put over' is not a person, but a thing. https://idioms.thefreedictionary.com/put+one+over+on+someone