Annie Clark shared some insights about the song in an interview with Pitchfork:

The song wasn’t intended as this moral indictment of the state of the pharmaceutical industry in the U.S. I don’t think it works to write finger-wagging songs, because it’s condescending to the audience and just a bummer to listen to. This song is super personal for me, a little snapshot of a small period of my life. I was having trouble sleeping and I had taken a sleeping pill. As I was popping it into my mouth, I was like, [sings] ‘Do-do-do, do-do-do, pills, pills, pills, every day of the week—oh, maybe that’s so jingle-y that it’s good.’ Just using that language of advertising.

In the same interview she also talked about the Eminem influence on the track:

I love that song [“The Real Slim Shady”]. I was definitely listening to Eminem, because that stuff still bangs so hard. This song is intended to be kind of a banger.

Annie Clark and producer Jack Antonoff discussed the song, saying:

He did indeed. Clark explained how she and Sounwave linked up in an interview with Pitchfork:

I’d worked with Sounwave on a cover of the Stones’ “Emotional Rescue” for a film called A Bigger Splash. We hit it off and worked on a couple of other tracks together. When Kendrick was making DAMN., Sounwave came to my studio, and I just freestyled on guitar for a long time on different ideas that Kendrick was working on. None of that stuff ended up getting used on the record, but Sounwave is always looking for the next, the most innovative thing.

St. Vincent and Cara Delevingne had been in a relationship for about a year and a half before breaking up in 2016. Not much is known as to why they split, but it seems to have been on good terms – Clark had told Vogue that they’ll, “always be really close. I’ll love her forever.”
As far as Delevingne’s significance to the song in perpendicular, it is very possible that the song details “pills” (whether a metaphor or literally) as being a problem they faced during their relationship, or the song could possibly be about a societal observation that they had made while still together. Of course, these are just guesses as to the deeper meaning of this track as Clark, herself, hasn’t yet spoken about it in detail.
It’s also worth noting that many St. Vincent fans speculate that the song “New York” is about the Clark/Delevinge break-up. In fact, it’s speculated that the whole record is somewhat based around this break-up.