There is a connection that is for sure, and its all a tad confusing due to the changes in clues that are given yet the Original Idea was more simple. But i wont go into much detail as i covered this in a reply earlier today.
Basically those Urns however do contain a Substance, that has a effect on Organic Life and Matter, One Series of Events from interaction with this Substance resulted in the Trilobite.
The Trilobites function indeed is the same as the Face Huger, they are related in that sense... the resulting Organism from the Trilobite was the Deacon, that indeed had a connection with the Mural Organism in Prometheus.
The Deacon however did not look too close to the Xenomorph, it was more Humanoid looking which i put down to being as it was born from a Human/Black Goo (Xeno-virus) Hybrid Face Huger.
The Neomorph indeed is similar to the Deacon, it is more flesh colored but it has a similar Head and Mouth and the Neomorph actually looked like it was more connected to Human DNA than the Deacon... It also had a Tail.
The Neomorph again is a Organism that came from the Black Goo so as the Black Goo seemed to contain some Genetic Parasite or Mutagen that contained Xeno-Related DNA or indeed Programs Genetic Material to take on Xeno-Related DNA
We have yet to have a real explanation for the Spores, but its indicated these had infected some Life on Paradise/Planet 4 and David saw and studied the results of this and he experimented upon them and also some Black Goo with various Organic Life and Matter in a attempt to experiment and create the Perfect Organism.
The Spores are seen in the movie at the edge of Running Water, the Juggernaught with its Cargo Bay open had crashed into the Mountain over what appears to be a stream of water that runs from the Top of this Mountain down to the River.
The open Cargo bay had acted like a scoup and scooped up some of the ground inside, and we can see Algae/Moss or other Plant Life had started to grow inside the Ships Cargo bay, and the Water from the Mountain is seen running under and near the Cargo Bay Entrance and this Plant Matter, and among this are the Spores that Hallet has inhaled.
Prior to this we see that Ledwood was located at the side of the River which the water running down from the Mountain would have ended up in... and by the side again we have Spores.
So its logical that at the side of this stream, we have algae, moss and other Vegetation and growing from the soil/vegetation was likely some kind of Fungus.
Fungus are not Technically Plants, they are Organic and so we can safely assume some Fungi/Spore like Organism had been infected with the Black Goo as it entered the Water from the Cargo Hold and Evolved the Fungi/Spores into what we see in Alien Covenant.
This is just how i saw it though and its not to say this is correct.