Is it correct English to write "John he is my husband," or, "Mary she went to the store?"

Is it correct English to write “John he is my husband,” or, “Mary she went to the store?”

I hear this construction frequently from TV and Radio Journalists. Is there a name for this construction?

As the sentences stand, the pronouns are normally redundant however, there are some circumstances where you would use it with some modifications in punctuation.

The first is in poetic usage where the extra pronoun is used either as a matter of form (this construction is common in folk music) or to preserve metre, e.g:

Mary she went to the store
For to meet with the man she adored
And she watched by the hour
While he measured out flour
And loitered around by the door
Fol a diddle de roll, etc.

Another common construction is in response to a question. For example:

Q: Who is this John you keep mentioning?
A: John? He's my husband.

So overall it is not how native English speakers would usually construct a sentence in casual conversation or writing but neither is it wrong nor particularly inelegant.

Yes, but you are punctuating the sentence improperly. To record this kind of phrase in English, we use the em-dash to set the appositive apart from the rest of the sentence. This gives:

• John— he's my husband— [rest of sentence].

• Mary— she went to the store— [rest of sentence].

We use the em-dash to separate the appositive here, as opposed to the more typical comma, because the appositive phrase is an independent clause. You could reword the sentences as follows to make the comma fit in.

• John, my husband, [rest of sentence].

• Mary, who went to the store, [rest of sentence].

That said, both formats are grammatically correct and indeed common in spoken English.

The emdash is often represented in written English in keyboard-recorded media via the digraph '--', rather than the more traditional character '—'

The construction is called "left-dislocation topicalization". Here's a paper¹ about the phenomenon, which defines the term in the introduction:

In this paper, I will investigate the information-structural properties of two English structures which utilize word-order variation for such purposes. (1a) and (1b) provide examples for the constructions.

(1) a. Tom, I like.  
    b. Tom, I like him.

The common name in the literature for the configuration in (1a) is "Topicalization" (abbreviated as TOP henceforth), while (1b) is most commonly called "Left-Dislocation" (abbreviated as LD).

^{[1]_{On English Topicalization and Left-Dislocation from an Information-Structural Perspective, by Péter Szűcs}}