This quote demonstrates that Mrs. Hardcastle is not romantic, and cares only for herself.
Mrs. Hardcastle wants Miss Neville (Constance) to marry Tony, and therefore she has kept Tony’s age a secret. If Constance marries Tony, Mrs. Hardcastle will have access to the jewels. When Mr. Hardcastel tells Tony he has been of age for three months, he can refuse Constance.
Witness all men by these presents, that I, Anthony Lumpkin, Esquire, of BLANK place, refuse you, Constantia Neville, spinster, of no place at all, for my true and lawful wife. So Constance Neville may marry whom she pleases, and Tony Lumpkin is his own man again. (Act 5)
When Mrs. Hardcastle talks about the “whining end of a modern novel” she is demonstrating her lack of romanticism and happy endings. She did not get a happy ending, because she wanted the jewels. The play does have a happy ending in that Constance is free to marry Hastings, and Tony does not have to marry Constance. Mrs. Hardcastle refers to Tony as " undutiful offspring" because he does not do what she wants him to do.