How is the supernatural involved in Samuel Coleridge's poems "The Rime of the Ancient Mariner" and "Kubla Khan"
According to Coleridge, he wrote Kubla Khan while under the influence of opium, although most critics believe that this claim, whether true or false, is inconsequential to appreciating the poem. However, because he was supposedly in a dream-like state when he wrote it, and he is writing about what he remembers about his dream, there are naturally supernatural elements involved since he cannot remember everything he dreamed and dreams in and of themselves are somewhat supernatural. Since you have not asked about the meaning, you can read about it here on eNotes. As far as supernatural elements, you need to look through the poem and find things that illustrate the supernatural. For example:
A savage place! As holy and enchanted
As e’er beneath the waning moon was haunted
By woman wailing for her Demon Lover!
The idea of a "demon lover" appears often in literature and the use of the word "enchanted" implies the supernatural.
And ‘mid this tumult, Kublai heard from far
Ancestral voices prophesying war!
If the voices are "ancestral" then they must be supernatural because the speaker's ancestors are dead. Continue on this way.
As for The Rhyme of the Ancient Mariner, you must follow the same plan. Go through it to find elements of the supernatural. For example:
At length did cross an Albatross,
Thorough the fog it came;
As it had been a Christian soul,
We hailed it in God's name.
The albatross appears out of nowhere, as if sent by God.
When the ancient mariner shoots the albatross, bad luck comes their way:
For all averred, I had killed the bird
That made the breeze to blow.
Ah wretch! said they, the bird to slay,
That made the breeze to blow
This is a supernatural element - implying in some way that the bird as a force of nature is getting his revenge on a supernatural plane of existence.
After this, there are many additional supernatural elements related to the shooting of the bird and the subsequent plight of the sailors:
About, about, in reel and rout
The death-fires danced at night;
The water, like a witch's oils,
Burnt green, and blue, and white.
You can take it from here, I believe. There is also an analysis of this poem here on eNotes.