Captain James Cook (1728-1779) was an English sailor renowned as a navigator, who is remembered as the discoverer of Australia (or at least the first Englishman to do so).
There are several mentions of Ned Kelly, an infamous bush ranger/ bandit/ folk hero in Australia and subject of several films - the earliest in1906 and the most recent in 2019.
The poem given to Smiley by his father is "The Dog's Meeting", a bush ballad of anonymous authorship.
The song Mrs Gaspen sings uses lyrics from the poem, "Cherry Ripe" by Robert Herrick.
The poem that the teacher expects Smiley to recite is "Horatius" by Thomas Babbington Macaulay, Lord Macaulay. (Note that Smiley also quoted from this poem in the earlier film, Smiley (1956)).