“Passer son temps à faire quelque chose”

Means spending a lot of time, sometimes unreasonable depending on context, on a particular activity.
i.e. :

“Il passe son temps à répondre à des questions sur Stack Overflow”

The only difference carried in :

“Avoir passé son temps à faire quelque chose”

Is the past tense variant (passé composé), meaning that the person has spent a significant amount of time, but no longer does it. The exact time period would then be specified by context.
In your example, the action (“relying on you”) seems to have been carried since the mother’s disappearance until present. Hence, the author wants to say he has been relying on his relative but wants it to stop now.
The question of the time spent being unreasonable, decent, reasonable, purposeful or not etc… is given by context.

However, we often use this expression when we want to stress that the action carried out is inappropriate or useless, or that the time spent is unreasonable.

“Avoir passé” is something that has been done in the past.

“Passer” is something that can happen nowadays or can be a habit.