~have the least weight with me

The narrator recalls his childhood.
He lives with his mother(Clara), Peggotty the only maid of his house, his stern stepfather Mr. Murdstone(Edward), and Mr. Mudstone's eccentric elder sister(Jane) in his late father's house.
His mother just now complained to her husband and his sister about excluding her from household economy.
And then, Miss Mudstone said he would leave tomorrow and Mr. Mudstone grumbled to Clara about her complaint.

"Don't, my love, say that!" implored my mother very piteously. "Oh, don't Edward! I can't bear to hear it. Whatever I am, I am affectionate. I know I am affectionate. I wouldn't say it if I wasn't certain that I am. Ask Peggotty. I am sure she'll tell you I'm affectionate."
"There is no extent of mere weakness, Clara," said Mr. Murdstone in reply, "that can have the least weight with me. You lose breath."
[David Copperfield by Charles Dickens]
I'd like to know what "have the least weight with me" means.
Thank you in advance for your help.