What you and your wife did is a very nice thing, and to me it seems you did two very separate things. Firstly, you made regular donations to an organised charitable program that you trusted would use the funds benevolently. As you stated, you did not directly send money to any one child. But secondly, your wife took the time to write to a child which of course is the point of such "sponsorship" programs, but not required in order to make a charitable donation. I think it would be helpful in this situation to view these as two separate things.
Your wife is no longer communicating with a child through an organised charitable programme. She is now directly communicating with an adult. Their communications are no longer going through the charitable organisation's channels, so there is no filter. What the adult is now saying to your wife is likely to be more influenced by their own personal needs and those of their family. I don't want to be cynical here, but there is also a possibility that your wife is not even communicating with who she thinks she is.
To me the dilemma you are facing is similar to when you see a homeless person on the street asking for money. You could give them money directly, but what will they spend it on? Or you could support homelessness by donating to a charity that will definitely spend it on food and shelter for homeless people.
If you want to continue helping the same charitable cause financially, could you continue your donations, or find a charity that helps people of all ages in the same country? If that is the case, whenever your wife feels that they are directly or indirectly asking for money she could perhaps respond by saying something like:
I am sorry to hear of the difficult situations you are experiencing. I hope that the donations we are making to [name of charity] are continuing to help you all.
By doing this you are indicating (i) that you are still helping financially, and (ii) by naming the charity you donate to you are showing them where they can go to get help. Even if you cannot continue supporting it right now you could still name the charity and say that they are continuing to help.
While charities are not all perfect, and the administration of some of the larger ones can mean only a fraction of your donation actually goes to their cause, at least you can be assured (if the charity is respected and transparent) that their funds are being spent on what you expect them to be, and that they are distributed through organised programmes.
If you are struggling with the decision of whether your money will go further when sent directly or through a charity, a helpful resource may be either Charity Navigator, or Charity Clarity in the UK, who both rate charities by their accountability and transparency.