Two places for records of birth for Corkwww.irishgenealogy.ie
and an Irish History Foundationwww.rootsireland.ie
Burns is anglo-irish spelling and Byrne is Irish.
Women disappear in records due to marriages.
Was Jane in Ireland too or the UK, where were the parents James Burns/Jane Topping going for James to have been born at sea, as it doesn't sound as though they were all headed to Auz?
Is there an early census in the UK? The earliest the birth records for Jane Toppings on www.freebmd.org.uk
and marriages from 1837.
Tithe Applots. 1823-1837
4 Byrnes, Parish's Churchtown, Dunmahon, Glanworth, Kilnamona and 21 Burns in various parisheswww.askaboutireland.ie
Griffith Valuation 1847-1864.
Looking to what names he gave his children for clues to his parents names. That might help.