I sponsor children, and haven't heard anything bad about the organisations, but I can't vouch for them 100%. There will always be corruption and abuses of power even if the vast majority of people in an organisation are completely decent.
The organisations I sponsor with are:
ActionAid. They are the most political (left-wing). They got some stick on here not long ago because one of their employees told someone they don't believe in biological sex. Since then I've been reading more of their material and I can tell you that doesn't represent ActionAid. They're well aware of how girls are disadvantaged because of their biology.
Plan International. Another secular organisation that puts a lot of focus on girls' rights. Again I haven't seen any sparkle-type ideology from them.
SOS Children's Villages. You'd probably be one of many sponsors for the child. These are children who can't live with their parents, so are part of a family in a village with an 'SOS mom'.
World Vision. A Christian charity who are a bit on the 'hard sell' side. They seem okay but have the highest turnover in the sense that you'll sponsor a child for a bit, then out of the blue you'll get a letter saying 'so-and-so has left the project, please meet so-and-so'.
Compassion. An evangelical Christian charity and the most old school. All the rest, you actually sponsor the project, not the child. But Compassion uses the old model which means your money goes directly to the child and their family. You can also sponsor them through college, and you can give fairly large monetary gifts apart from the monthly donations.
All of these charities are good in their own way. When people ask I tend to recommend Plan as having a good reputation and being politically aware and committed to social change, but also secular.