People generally don't like having their immediate needs exploited by those seeking to proselytize an ideology. I'm not sure that's an "issue" that needs to be "worked on". If you're being altruistic and want to help people in need, the help you're offering is about them, not you -- if you make it about you, why would you be surprised when people doubt your intentions?
Imagine if the local library were destroyed, and a group came forward to organize book donations, but due to the religious beliefs of the people running the book drive, they only were willing to collect evangelical literature, and refused everything else. Would the locals in that scenario be wrong to avoid them?
The issue is deeply cultural & until it's being worked on, solidarity is key: serving people in the ways they choose to receive care.