Is it 'Have a Cold' or 'Have Cold'?

It’s “have a cold”. When referring to a mild, very common illness, the noun cold is countable, and its singular form needs an article.

Have a Backache or Have Backache

You can use the expressions “have a backache” or “experience a backache” in American English to refer to a continuous pain in your back.

Have a toothache vs. have toothache

In American English, we normally use the expression “have a toothache” when experiencing a pain in or about a tooth.

'On Holiday', 'On a Holiday', or 'For a Holiday'

In British English, you can use the expression “on holiday” to mention a period of time away from school or work. This is the equivalent of the American “on vacation”.