Possibly Dear Devil (1950) by Eric Frank Russell, also the answer to Anthology for children; contains story about friendly Martian on postapocalyptic Earth. The story is available at the Internet Archive. (link snatched from the answer there, quotes OCRed from the Archive link)
One of the aliens, who they call a 'poet' decides they wish to stay behind and the captain of the aliens reluctantly lets them go.
That poet would be Fander.
"Same with this." Fander exhibited his rough sketch. "This holds beauty. Where there is beauty there once was talent—may still be talent for all we know. Where talent abides is also greatness. In the realms of greatness we may find powerful friends. We need such friends."
"You win." Skhiva made a gesture of defeat. "We leave you to your self-chosen fate in the morning."
"Thank you, Captain."
The poet then goes on to create a home, and eventually finds a human child who they look over and nurse back to health.
A group of children, actually, one he does nurses over.
On the seventieth day, in a deep, shadowed glade to the north, he spotted a small group of new shapes slinking along in single file. He recognized them at a glance, knew them so well that his searching eyes sent an immediate thrill of triumph into his mind. They were ragged, dirty and no more than half grown, but the thing of beauty had told him what they were. [...]
This one was dark-haired, a bit bigger, and sturdier. It fought madly at his holding limbs while he gained altitude. Then suddenly, realizing the queer nature of its bonds, it writhed around and looked straight at him. The result was unexpected: it closed its eyes and went completely limp. [...]
It was still limp when be bore it into the cave, but its heart continued to beat and its lungs to draw. Laying it carefully on the softness of his bed, he moved to the cave's entrance and waited for it to recover. Eventually it stirred, sat up, gazed confusedly at the facing wall. Its black eyes moved slowly around, taking in the surroundings. Then they saw Fander. They widened tremendously and their owner began to make high-pitched, unpleasant noises as it tried to back away through the solid wall. It screamed so much, in one rising throb after another, that Fander slithered out of the cave, right out of sight, and sat in the cold winds until the noises had died down.
They eventually meet an older human who threatens them with a gun before agreeing to assist them.
Meet lovely "elder" Graypate:
The sled climbed, glided a mile at five hundred feet. Fander's atten-tion was divided between his limp prizes, the controls and the horizon when suddenly a thunderous rattling sounded on the metal base of the sled, the entire framework shuddered, a strip of metal flew from its leading edge and things made whining sounds toward the clouds.
"Old Graypate," bawled Speedy, jigging around but keeping away from the rim. "He's shooting at us."
The story ends with the original martians coming back and finding the settlement, now larger, and talking about the poet.
Fander has been "dead" for a while, but isn't forgotten:
Weep, my country, for your sons asleep, Tire ashes of your homes, your tottering towers. Weep, my country, 0, my comely, weep! For birds that cannot sing, for van-ished flowers, The end of everything, The silenced hours. Weep! my country.
There was no signature. Rdina mulled it through many minutes while the others remained passive. Then he turned to Speedy, pointed to the Martian script.
"Who wrote this?"
"One of your people. He is dead." "Ah!" said Rdina. "That songbird of Skhiva's. I have forgotten his name. I doubt whether many remembered it. He was only a very small poet. How did he die?"
Actually, the process he went into is kind of a sleep he could be woken from by the Martians.
Found by searching this site for
[story-identification] poet mars which returned the linked question.