Irony is hard to define. It is usually something like a joke that would be funny if it were not too serious, too painful, or too tragic. The only really strong element of irony in Saki's story does not occur until the very end.
Gortsby has listened patiently to the story the young man tells him about losing his hotel and needing a loan in order to rent a room at some other hotel. Gortsby is evidently accustomed to sitting on park benches, and he probably does this around dusk because he works in an office somewhere and wants to enjoy the outside air for a little while before going to his room or rooms. He is described as relatively secure financially but certainly not rich. He is a little better than an ordinary clerk but not a professional, more likely some sort of office manager or store manager. So his home is probably quite modest. It might be what the English used to call a "bed-sitter," a furnished room with some cooking facilities. It would be understandable that he wouldn't be in any hurry to get back to such a place.
Gortsby has heard plenty of hard-luck stories sitting on park benches. There is no better place to attract strangers with hard-luck stories than a bench in a city park. He probably has no intention of giving the young man money regardless of whether or not he is telling the truth. The stranger is encouraged by the fact that Gortsby is listening. At last he says,
"I'm glad, anyhow, that you don't think the story outrageously improbable."
He threw a good deal of warmth into the last remark, as though perhaps to indicate his hope that Gortsby did not fall far short of the requisite decency.
"Of course," said Gortsby slowly, "the weak point in your story is that you can't produce the soap."
Up to this point there has been no irony. The young con man leaves in a huff. Then Gortsby finds a cake of soap on the ground by the side of the bench. He is overcome with remorse and goes chasing after the stranger in order to lend him a sovereign and give him the soap. Gortsby feels ashamed of himself for having had such a negative attitude towards his fellow man. He tells himself:
"It's a lesson to me not to be too clever in judging by circumstances."
But when he passes by the bench where he had been sitting, he sees an elderly gentleman "poking and peering beneath it and on all sides of it."
Gortsby recognizes him as the man who had been sitting beside him before the young stranger took his place.
"Have you lost anything, sir?" he asked.
"Yes, sir, a cake of soap."
This is where the irony comes in. Not only does Gortsby realize he has lost money and been made a fool of, but the reader experiences similar feelings because he has been sharing Gortsby's impressions and feelings all along. The reader thought the young man was a con artist. Then the reader felt he had to be telling the truth because the cake of soap seemed to prove the veracity of the hard-luck story. And when it turns out that the cake of soap was dropped by the elderly gentleman, the reader emphasizes with Gortsby's pain, which is a necessary component of irony. Gortsby feels angry, humiliated, and poorer by a sovereign, which for many clerks was a week's wages.
Not only that, but now Gortsby has no idea whether or not the elderly gentleman is another con artist who dropped the soap on purpose before leaving, intending to come back to look for it and use it as an excuse to start a conversation with Gortsby, whom he expected to find still sitting there.
What's commection with hope
The only connection with "hope" that I can think of offhand is that when Gortsby becomes convinced that the stranger is really a young aristocrat from the country, Gortsby rushes after him to give him some money seemingly in the hope that he can make a friend of a member of a higher social class. The tale told by the young con artist is designed to make Gortsby think this is a real possibility. The young man is a complete stranger in London. He might be happy to make the acquaintance of a knowledgeable Londoner of his own age who would show him around town and give him some practical advice. In return, as Gortsby hopes, his newfound friend might invite him down to his country home for shooting, riding, and meeting upper-class country people. Gortsby's hope is never expressed in so many words, and such a friendly relationship is never suggested by the stranger; but this would seem to explain why Gortsby is so anxious to apologize and make amends for his discourtesy by "lending" the young man a sovereign. Gortsby feels he has everything to gain and nothing to lose, since he will surely get his sovereign back in the mail within a couple of days.