Learn English – When to use potentials in a sentence

Traditional grammar defines the sentence as 'a complete thought containing a subject and a predicate'. So by this definition the second 'sentence' is incorrect, but of course writers constantly break the rules of grammar to achieve a particular effect.

The problem I see with the 'sentence' is that it may lead the reader astray. He or she gets to the end of the string of words (a long noun phrase) thinking that they comprise the subject, but then the predicate is missing. This will possibly cause a backtrack to reread the first sentence in order to make sense of the whole. It's generally inconsiderate of your readers to make them do this.

(And no, I would not recommend breaking this traditional rule in an application letter.)

There is no fault with using wonder + conjunction, like in

I wonder why he did that.

I wonder where he is/went.

My concern is elsewhere: what you mean with "where" is "what place" ("he" wants a name) or "which place" ("he" already knows a list of places and he wants to know the one).

I believe that:

Mike really wonders where you visited in England last week.

is not necessarily wrong, but it would perhaps leave the person wondering for a split second if they were hearing it, and frown slightly if they were reading it. The problem is that visit introduces an accusative object (one visits a place), not an adverb + location (one is in a place). That is why I would see the construction where you visited as slightly abusive.

So it would be more appropriate (as well as precise) if you said:

Mike wonders what/which place(s) you visited in England last week.

But it remains a matter of taste and usage. Perhaps there are regions where your version is perfectly idiomatic in spoken language; and no doubt eminent writers already used it?