"Although they work in most cases, they cannot handle cases when a comment or script is broken by the cutting"
Should I put "the" between "handle" and "cases"?
I think both constructions are similarly clear to the reader. For some people, I think there is a slight difference in connotation depending on what you mean by "most cases" in your opening clause.
If what you are referencing works in all cases except those in which a comment or script is broken by the cutting, then using "the" seems more appropriate to me. If what you are referencing works in most cases but doesn't work in some cases, including but not limited to those in which a comment or script is broken by the cutting, then dropping "the" seems seems more appropriate to me.
"The" is a definite article and usually is used to refer to something specific. In this case, for some, using the article "the" will emphasize the uniqueness of the cases that you mention. In the first example, the cases that you mention are particularly unique. Including a "the" makes this more apparent. In the second example, the cases that you mention are different but not particularly unique. Dropping the "the" de-emphasizes the uniqueness and might help some people to realize what you are referencing might not work in other scenarios as well.
It's not particularly relevant that you have noun followed by a clause. For the purposes of this discussion, just treat the whole thing ("cases when a comment or script is broken by the cutting") as an object.
[Post edited to better deserve it's upvote!]
"The" is a definite article. So the only reason you might to add "the" is if you have mentioned "broken comments or scripts by the cutting" earlier in your writing.
The sub-phrase "the cutting" seems odd, but we can assume that "the cutting" is some domain-specific term or a qualified noun (e.g. with an implied adjective) in a larger context.
So let's look at the core flow:
When spoken it sounds awkward. That's because it has a plurality conflict:
This is because of the sub-phrase
has a singular subject.
Although they work in most cases, they cannot handle the case when a comment or a script is broken by (the) cutting
To clarify let's consider a slightly simpler phrase:
Please note that even though the compound phrase is singular, the phrase does not lose the sense of cardinality because the entire noun-phrase becomes a common noun.
Now for the the in the cutting and whether it's a qualified noun:
However if in a previous context "the colour red" was qualified you can drop "red":
Applying this approach to your phrase then:
Proper construction would be either: