What did Mr. Bumble want Mrs. Corney to agree to do in Charles Dickens's Oliver Twist?

The short answer is, marry him. Chapters 23 and 27 of Oliver Twist are concerned with the courtship of Mr. Bumble and Mrs. Corney. It is a very brief courtship and many things remain unspoken.
When Mr. Bumble comes to visit Mrs. Corney (who has been a widow for 25 years) in her rooms he gradually draws his chair closer to hers and then kisses her. She protests but not very convincingly. She is then called away to tend to Sally, who is at death’s door. While she is gone, Bumble takes an inventory of the room and apparently judges its quality worthy.
In the later chapter, when she returns, he manages to make his intentions clear without actually asking the question. Bumble points out that Mr. Slout, the manager, is expected to die soon, while will leave a “wacancy.” He implies that if the two of them are married, they could run the house together, calling it an “opportunity for a jining of hearts and housekeepings!” His proposal is simply to ask her to say “The little word? . . . The one little, little, little word . . . ” The “bashful beauty” stammers out “yes.”
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