A similar question was asked here, but I'd like to add a few new examples and am seeking clarification.

In most scenarios, it sounds natural to say "in the 1st/2nd/3rd/4th week of a month". But below are a few examples where the statements use "on the 1st/2nd/3rd/4th week of the month". Are they exceptions (cannot see why they are) or a misuse of the preposition ?

Example 1

As required by the EPA and the New Mexico Environment Department, the Town of Silver City is preparing to perform lead and copper testing of its drinking water on the fourth week of July 2017 (July 24 – July 28).


Example 2

This schedule proposes holding the departmental budget hearing on the fourth week of July


Example 3

We held a BBQ on the second week of July



First, I think you're correct when you say that in would be a better preposition than on for these examples. As one commenter said, on is typically used when referring to a specific day (e.g., on Monday, on the last day of the quarter, on her birthday, on the thirteenth).

That said, I noticed that none of your examples were written by authors of published books, or by journalists writing for newspapers. Instead, they look like announcements, perhaps cobbled together by someone taking notes at a meeting. So it's not too surprising that this minor mistake was made, and it's the kind of thing most native speakers wouldn't notice too much.

When the event is relatively short compared to the week itself, and the specific date is not yet known, you might use during or in, and qualify that with sometime:

We will hold the departmental budget hearing sometime during the fourth week of July.

The company picnic will be held sometime in the third week of August.

If the event lasts multiple days, such that it spans all or most of the week, then we could use during or throughout:

The town is preparing to perform testing of its drinking water during the fourth week of July.

Tryouts will be held throughout the third week of August.

Note that throughout in the second set of examples suggests there are many days of tryouts, not just one. If we used during instead, that could mean many tryout sessions over the course of the week, but there might also be just one day of tryouts sometime that week.

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