Def 1: some one that cannot assert themselves when necessary
Def 2: a problem an infant has when he/she cannot learn to breastfeed/suckle
Ex 1: "Mark said that guy was a weak suck because he was letting his ex-girlfriend walk all over him."
Ex 2: "The newborn could not get the nutrients needed due to the fact she had a weak suck."
Someone or something that is weak in their efforts and suck at doing it.
That soccer player is a weak suck. He doesn't try and he looks bad doing it.