The passage is an example of free indirect speech. The narrator renders Mr Plornish's speech without quotation marks but he does not use indirect speech either.
Using direct speech, the passage may have read as follows (without transcribing it completely):
Mr Plornish said, "Then you see, some people as was better off said that they was ‘improvident’ down the Yard. For instance, if they see a man with his wife and children going to Hampton Court in a Wan, perhaps once in a year, they says, ‘Hallo! I thought you was poor, my improvident friend!’ Why, Lord, how hard it was upon a man! What was a man to do? (…)
In this version, I have added the quotation marks and removed a few comments that probably come from the narrator. The comment "that was the favourite word" definitely comes from the narrator. The comment "and a good many such people lived pretty close up to the mark themselves if not beyond it so he’d heerd" probably also comes from the narrator.
Using indirect speech, the passage may have read as follows (without transcribing it completely):
Mr Plornish said that some people who were better off confessed that they were improvident (…) He wondered what a man was to do.
Free indirect speech allows the author to omit (or reduce) the use of reporting clauses ("he said", "he asked", "he wondered", …) and to present a character's speech or thoughts without adding the quotation marks. (The only quotation marks in the passage are used when Mr. Plornish quotes another person's words.)
"He couldn’t go mollancholy mad" refers to any man who was "improvident" (to use Mr. Plornish's favourite word) and who had lost hope ("What was a man to do?").