The best summary of Adcock's "Immigrant" is the one she gives herself that is printed (with audio) with Adcock's permission on PoetryArchive.org. Adcock, using a very apt and interesting double metaphor built on swans and pelicans in St. James' Park, says the poem is a retrospective, a look back at her feelings eight months after immigrating from New Zealand to England. She wanted very much to fit in as a "genuine Londoner," as did all other immigrants, and one of her first efforts toward fitting in was to begin to lose her New Zealand accent.
In the poem, looking down from a low bridge over the St. James Park lake, she contemplates the swans swimming on the lake next to the pelicans that swim alongside. Pelicans are avian immigrants to England, like Adcock herself, while swans are native to England, which Adcock is not. She compares herself to the pelicans who do a fair job of looking like the swans but who, knowing this themselves, are only "slightly ruffled ... awkward" representations of what they would like to be, swan for the pelicans and Londoner for Adcock.
... watch the pelicans:
they float swanlike, arching their white necks
over only slightly ruffled bundles of wings,
burying awkward beaks in the lake's water.
To reveal her own slight ruffling and awkward feelings, Adcock describes "testing" her accent again: does she sound like a Londoner when she says "St James's Park" to herself ... or ... does she sound like a new Zealander?
St James's Park; St James's Park; St James's Park.