Little Dorrit has told Arthur Clennam a few lines previously that she has a third thing to to say to him and after extracting Clennam's promise not to think her unreasonable or ungrateful, she says
You are coming back to see my father again?
and then
You have been so good and thoughtful as to write him a note, saying that you are coming to-morrow?
So it is clear that she is telling Clennam not to encourage her father to make some request of him. If Little Dorrit and her family really need that thing, she will ask for it and Clennam is to discourage her father from asking, to pretend not to hear him if she does ask.
My understanding is that she is asking Clennam not to tender any further 'testimonials' (doles of cash) to Mr Dorrit. She understands that he has paid out money on her brother's account, but does not wish him to keep paying out monies to her father as he does not put it towards resolving his debt, but to maintaining his social prison as Father of the Marshalsea.
Back in Chapter 8, Clennam is giving Richard Dorrit money:
Mr Clennam had two things to do before he followed; one, to offer his testimonial to the Father of the Marshalsea, without giving pain to his child; the other to say something to that child, though it were but a word, in explanation of his having come there.
But by the time we get to Chapter 22 he has ceased:
His obtuseness on the great Testimonial question was not calculated to awaken admiration in the paternal breast [...] Howbeit, the father did not fail in any outward show of politeness, but, on the contrary, honoured him with much attention; perhaps cherishing the hope that, although not a man of a sufficiently brilliant and spontaneous turn of mind to repeat his former testimonial unsolicited, it might still be within the compass of his nature to bear the part of a responsive gentleman, in any correspondence that way tending.
Basically this is saying that Clennam has not volunteered any more cash gifts to Little Dorrit's Father, but that Richart Dorrit still hopes that he will give him some money if appealed to by letter. Clennam receives a letter from him in which Little Dorrit's father mentions that:
most unexpectedly finding himself in the novel position of having been disappointed of a remittance from the City on which he had confidently counted, he took up his pen, being restrained by the unhappy circumstance of his incarceration during three-and-twenty years (doubly underlined), from coming himself, as he would otherwise certainly have done—took up his pen to entreat Mr Clennam to advance him the sum of Three Pounds Ten Shillings upon his I.O.U.
And for the avoidance of any doubt, after Clennam has responded to this letter by paying the requested sum he says to himself, while thinking of Little Dorrit:
When she had seen her father begging with his threadbare disguise on, when she had entreated him not to give her father money...
