When Kara is pleading to the Argo council, behind her are two sets of two rings. They are identical to the spinning rings which held Zod, Ursa, and Non during the trial scene in Superman: The Movie. The two scenes are thematically similar.
The little boy that Mon-El heals in Argo City is named Val. Though not directly related to Supergirl (2015) in and of itself, Kara's great-great grandfather, Val-El, appears in the series Krypton (2018).
Argo City also appeared in Supergirl (1984), and is from where Kara arrived on Earth in search of the Omegahedron.
When Selena is given the vote to break the tie if Argo City shall help Earth, it is similar to when Jor-El is given the last vote to put Zod in the Phantom Zone.
Dark side of the moon is mentioned in the song Kryptonite