The most common and most likely answer to this crossword clue is the 6 letter word SORROW
S
O
R
R
O
W
More clues leading to the result SORROW
- Briefly hurtful argument causing grief
- Regret a lot of vexed argument
- In this manner Romeo’s right to express pain and grief
- Basis for wholesaling line creates grief
- Feeling of grief in this way starts never-ending quarrel
- Angry, endless argument causing pain in the heart
- Regret, likewise, start of rancorous dispute
- Start of squabble or argument making us feel sad
- Class mostly argue, giving one grief
- Thus resistance precedes quarrel and sadness
Buddy explains!

The straight part
We know that "that could make you weep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that could make you weep = sorrow.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SORROW mean?
SORROW noun- Unhappiness, woe
- (usually in plural) An instance or cause of unhappiness.
- To feel or express grief.
- To feel grief over; to mourn, regret.