Where should we use 'half a pound' and where 'half pound'?

We use half/half of/a half of pound before a noun with a determiner( the, my, this, that, who, when etc).Half or half pound if followed by plural noun, also takes plural verb.

Half a pound
In grammar half usually comes before the article a/an and the, but it is quite possible to put it after in the expression of measurement.
Example:
I have bought half a pound of fish.
Or
I have bought half a pound fish.
Notice that i skiped of in latter sentence which is also correct.

Now come to half pound:
It is also correct to use, if am not wrong Americans would rather prefer to skip a from such sentences( not sure).

Last and most important tip, which you didn't ask is A half pound
When we usually talks about the point accuracy we use a half pound.
Example:
I purchased two and a half pound fish.
Kept this thing in your mind that in case of skipping "a"--indefinite article, your sentence is grammatically incorrect.
Like it could be incorrect to say: I purchased two and half pound fish.

I hope my explanation has resolved your query.