How were the Romantic writers influenced by the Greeks? Are there any sonnets that illustrate this?
Many of the Romantic writers were influenced by images of Greek culture. Some, such as Byron, actually travelled to Greece and became involved in the movement for Greek independence from Turkey, and others were inspired either by reading the classics or seeing famous Greek artifacts such as the Elgin marbles in British museums. They described many of these experiences in their poetry.
A famous Romantic sonnet, John Keats' "On first looking into Chapman's Homer" was composed in response to Keats' reading of Chapman's translation of Homer's Iliad. Another poem by Keats, "Ode on a Grecian Urn" actually describes the "Elgin marbles", sections of the Parthenon frieze that Lord Elgin removed from the Parthenon and brought back to Britain, whether they are displayed in the British Museum.
Shelley was an atheist who saw in the Greeks a rational alternative to Christianity. Keats and Coleridge saw the natural experession of the Greeks and fluid structure of the ode as an antidote to the rigid forms of the Augustans, and especially to the heroic couplet.