What can make a well-intentioned prayer ministry weak or ineffective?
When we go before God on behalf of someone else and pray for what we want for the other person from a position of judgment, we short circuit the benefit of prayer. Effective prayer is seeking from God only that which is in the best interest of the other person(s) and God's will for them. Prayer is not a time for the pray-er to serve as judge and seek their own desires in the lives of others.
Put another way, motives matter in prayer. Look closely at James 4:2b-4, "... Yet you don’t have what you want because you don’t ask God for it. 3 And even when you ask, you don’t get it because your motives are all wrong—you want only what will give you pleasure."
Before seeking God on behalf of someone else, the first thing the pray-er should do is check their own motives. If you can't petition God from a non-judgemental position, then you first need to work out that problem with God's help before you bring requests on behalf of others.