I'm not quite sure what input I can provide on this subject... but I'll have a go

With regards to the Turkish woman knocking on doors, saying she is a neighbour and needs money for milk for her child,... well I think that this is a scam.
I do know that money is tight at this time of year, for Turkish people, and probably ex pats too.
Jobs are quite plentiful in the season, but in winter months, (especially if you work within the tourism industry), then jobs are quite hard to come by.
A typical Turkish family,(eg;husband at work, wife at home), may earn around 600TL per month. So that alone makes it very difficult to live. Prices have gone up on everything here, even fruit and veg.
Baby powder milk is expensive in Turkey, but that is not an excuse to go door to door asking for money.
As Hoopsman said there are charities that can be contributed to, and like him I never give money to a beggar. If they say they need bread.. I will buy a loaf and give it to them.
Even after all the years of tourism there is (sadly) still the odd person that thinks ex pats are full of money (or at least, better off than them), and they try it on. It has happened to me, even though I work for money to live, am a single parent with no help from my childs father, and no benefits from the Government.
People ask me about the prospect of Turkey joining the EU, my answer is always the same... if that means that they start to provide benefits then so be it! I can remember when Celal was 1 year old, I had just divorced, and returned to work. I was repping in Fethiye, an English couple had 'a go at me' because I was working, and my child was with a childminder, they said that I should be at home with my son and be claiming. Then I told them that unfortunately there aren't any government benefits in Turkey! This is what needs to change.
I have found it hard living here on a standard wage.... but have never begged.
So, although you will meet people that live hand to mouth, it is hard for them.. but begging isn't the answer... and I for one wouldn't give in to that. I'd find another way to try to help.
This is my personal experience and opinion
