Col,
Unfortunately, a lot of what you describe as occurring in the UK also occurs in the USA but I don't have ready statistics for it. Yes, there are the "professional" homeless who will unabashedly reveal that they beg because they get more than they would get on a day job and it is "off the books" i.e. not subject to tax withholding. And probably is also totally not reported or is severely underreported.
We have those charities that end up having over 66% of their haul going towards support of a profligate lifestyle by the charity's executives, though there ARE some sites that rate the efficiency of given charities. My wife and I donate after consulting such sites. If they don't put at least 75% towards their focus group then they don't get a dime from us.
To be honest, I don't think the wall will cut down that much on homelessness because I am under the impression (possibly incorrect but it is what I have heard) that the illegals get together and pool their resources so that they have a roof over their heads but they probably exceed any reasonable level of the number of families living together. What I have heard is that a very large percentage of our "true" homeless people are borderline psychotic or at least extremely neurotic. I.e. to be a bit indelicate about it, mental cases.
Then again, I was not in favor of the wall anyway. I favor border control, yes. I favor action on illegal immigration. But I feel that the wall is the wrong answer. Static walls have not been effective throughout history. What makes them effective is the number of people patrolling the wall. Reduce that and you open the floodgates. If we are not going to increase the number of border patrol agents, then I think the wall is going to be a monument to governmental waste.