"Sheriff's hiring of political supporters under fire" & "building a community outreach wing his critics say doubles as a re-election team"

Question

What's the grammatical structure of these:

1. Sheriff's hiring of political supporters under fire

is the title of a story. Who or what is "under fire"? Sheriff, supporters, or hiring?

2. Sheriff Scott Israel has hired from the ranks of his political supporters, building a community outreach wing his critics say doubles as a re-election team.

It's a little confusing. Should there be a "that" or "which" before "his critics", because "that/which" acts as both the object of "say" and the subject of "doubles"?

Answer 1

The first is an example of "headline English", about which you will find many questions on this site. Headline English omits as many unnecessary words as possible, leaving only the bare meaning. In some cases you have to understand the context to get the meaning, but in this sentence, imagine it was written as a complete sentence

[The] Sheriff's hiring of [his] political supporters [has come] under fire

Then, figure out the subject of the sentence, which in this case is the gerund "hiring". "Sheriff's" is an adjective saying whose hiring it was, and "of his political supporters" is just a phrase telling us who was hired. So it's the act of hiring that is the problem.

Another example:

'Mystery meat' servings in school cafeteria proclaimed 'inedible' by students.

Again, what exactly have the students proclaimed "inedible"? The school, the cafeteria, the servings, the meat, the mystery? Obviously not some of these, but, again, think of this as a complete sentence

Servings (of 'mystery meat') (in the school cafeteria) have been called "inedible" (by students).

This should make it clear that the subject is "servings of mystery meat".

Your second question should probably be asked in a separate post, but the short answer is that "that/which" is often optional. It's likely many English learners are told it is required, so they become familiar with the structure, but once you read more you'll find that native speakers take all kinds of shortcuts. Examples:

Please bring me the book (that) you bought for me last Christmas.

Sally goes to the same school (that) her mother went to as a girl.

Answer 2

In your headline

Sheriff's hiring of political supporters under fire

the act of "hiring of political supporters" is what is being criticized.

In your second example, the conjunctions "that" or "which" are optional and the meaning does not change with or without them.