Jewish people are very respectful of Abraham and would hesitate to use the word "weakness." Abraham passed all of the tests of faith with which God tried him (Pirkei Avot ch.5).
To answer your question, the only issue I can think of was when Sarah correctly told him to send Ishmael away, and Abraham hesitated until God reiterated her feelings (Genesis 21:9-12).
(Other things, such as when Abraham pressed God to give him children and concerning the inheritance of the Holy Land [Genesis ch.15], were actually motivated by his concern that his beliefs might eventually die out after his passing, unless a firm chain of tradition would be established. These cannot be considered weaknesses.)
Abrahams challenges were that he had to protect his kids.
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