cockup

[yuanmingqing]

Does cockup means making a mess of a job? Is it rude to use because of the word cock? Kindly cite me some examples to help me understand it.

[5jj]

Does "cock-up" means "making a mess of a job?" It can do.
Is it rude to use because of the word "cock?". Some people prefer not to use this BrE expression, but many happily use it informally.
Kindly cite me some examples to help me understand it.

There was a cock-up over the arrangements for the dance; we haven't got a band!

Kindly can sound a little demanding. It's safer to use please.

[amigos4]

Does cockup means making a mess of a job? Is it rude to use because of the word cock? Kindly cite me some examples to help me understand it.

I have never heard "cockup" used in AmE. We would say something is "screwed-up" or "fu_ _ ed up".

Anytime the word "cock" is used (when not referring to poultry) in AmE it will raise eyebrows! Its use in most instances would be considered rude.

[Tdol]

It's not rude in BrE- it's slang, but not the same as cock alone.

[Barb_D]

It would definitely raise eyebrows, as said above, in the U.S. We seem quite afraid of this word. It feels vaguely pornographic. While the f-word is still considered very vulgar by many and impolite by just about all, it's ubiquitous. You'd cause less of a ruckus saying "f'ed up" than you would "cocked up" here.