So a grammar site I look up things regularly states that this is a participle and that this construction is formal and not used often. Why can't it be a gerund? And aren't with + ing constructions used often? Because I see them a lot.
'ing' appears a lot in English, and it takes on multiple forms: words that end with ing can be a noun, verb, or adjective depending on how they're used.
This site has a good explanation of gerunds vs. present participles, but here's my version of the difference:
When you have a phrase like the "is eating" in "Jane is eating," this is a present participle. Basically, this is used to convey that Jane is, right now, eating. Present participles are verbs and are giving the "action" of the sentence. (This is different from some romance languages, for example French, where one verb form equates to the three English sentences, "Jane eats," "Jane is eating," and "Jane does eat." [in French: "Jane mange"]).
For example, these sentences all have present participles:
- I am eating bread.
- You are drinking coffee.
- Bill is doing his homework.
This is different from a gerund, which is where the 'ing' form is used as a noun - that is, it's the subject of the sentence, the object of a preposition, or a direct object. Some examples:
- Swimming is fun. [Swimming is the subject]
- I like walking. [walking is a direct object]
- I can't exercise without singing. [singing is the object of the preposition]
"With Mrs. Jones going to New York" is a participle phrase. This website explains it well, but the gist is this: A participle phrase modifies some other part of the sentence. You can always remove a participle phrase from a sentence, but you can't remove a gerund.
For instance: "Spending all day at the beach was fun, but I didn't do any of my homework." Ok, is spending a gerund or a partiicple phrase? Try taking it out of the sentence, and you get "was fun, but I didn't do any of my homework." The remainder makes no sense, so we know that this is a gerund (and "was" is the verb).
On the other hand: "Walking home from school, I saw a cool airplane." Take out the "ing" phrase: "I saw a cool airplane." Since we still have a sentence, "Walking home from school" is a participle phrase and acts as an adjective in this sentence (it modifies the subject, "I").
Regarding the formality of participle phrases, they certainly aren't informal (in the sense of colloquial, like "hey, what's up?"), but they also aren't so formal you'd never use it on a regular basis. I think you might be confused because your example sentence isn't entirely grammatically correct. First, the tense of the sentence is incorrect; I think that "With Mrs. Jones going to New York" implies some event that will happen in the future, so the second part of the sentence should use the future tense: "Mr. Smith will take up her position." Second, "will take up" is not the right expression for this; use "will take her place" or "will replace her": "Mr. Smith will take her place."
So the corrected sentence is "With Mrs. Jones going to New York, Mr. Smith will take her place [this week]." (The sentence probably has some kind of timeframe, whether it's written or implied.)
Honestly, for this sentence specifically, I probably wouldn't use a participle phrase and would instead use 'since': "Since Mrs. Jones is going to New York, Mr. Smith will take her place."
In other situations, a participle phrase is perfectly fine:
- Looking up at the sky, Jill saw two clouds and an airplane.
- Carefully searching each room of his house, Bill finally found his lost keys.
- Watching as Jill sped down the highway, the police officer turned on his lights.
Basically, participle phrases that start a sentence will work best when they describe the cause of a situation ("looking up," "carefully searching", "watching") and the remainder of the sentence describes the result ("Jill saw," "Bill found", "officer turned on his lights").