i didnt look for Mc Morne or OMorne, but did find a La morney by accicident
below [6 counties] from 1823-1837 Tithes Applotment Books based upon the index work of the Irish Genealogy Hub’s listing for each parish in Northern Ireland, The relevant original books are in the Public Record Office of Northern Ireland (PRONI) Belfast
Morn, Thomas-Townland: Cah Year: 1825-Errigal-Derry
Morney, Patrick-T: Fireagh Y: 1830-Drumragh-Tyrone
Mourn, Patt.-Tl: Glen Yr: 1832-Enniskillen-Fermanagh
Mourne, James-T: Loy Gardens Y: 1826-Derryloran-Derry & Tyrone
Mourne, James-T: Loy Gardens Y: 1826-Derryloran-Tyrone
below [26 counties]from
http://titheapplotmentbooks.nationalarchives.ie/Morn Jno Giants Grand Rathronan Tipperary 1827
Morne John Rockfield Rathaspick Westmeath 1833
Morne James Carrigagh Rathaspick Westmeath 1833
Morne Francis Ballygarvey Rathaspick Westmeath 1833
Morne Thomas Rathowen Rathaspick Westmeath 1833
Morney Denis Thomastown Carbury Kildare 1834
Morney James Killrahan Dysartmoon Kilkenny 1831
Morney Wm Killrahan Dysartmoon Kilkenny 1831
Morney Thos Killrahan Dysartmoon Kilkenny 1831
Morney Terence Knocknaskagh Kilmolash Waterford 1827
Morney Patt Moveen East and Mulcashine Moyarta Clare 1827
Morney Mary Wolfestown Rathmore Kildare 1842
Morney John Aughlish Tuam Galway 1831
La Morney Wm Ballihalwick Fanlobbus Cork 1826
Mourne Patrick Laragh Derrylossary Wicklow 1831
Mourne James Tomriland Derrylossary Wicklow 1831
Mourne Thomas Ashtown Or Ballinafunshoge Derrylossary Wicklow 1831
Mourne Elizabeth Ashtown Or Ballinafunshoge Derrylossary Wicklow 1831
Mourne Thomas Ashtown Or Ballinafunshoge Derrylossary Wicklow 1831
Mourne Thomas Toghermore Derrylossary Wicklow 1831
Mourne Daniel Rathoonagh Kilbride Mayo 1834
Mourne Anthony Crott Kilbride Mayo 1834
Mourne Anthony Crott Kilbride Mayo 1834
Mourne Daniel Rahoonagh Kilbride Mayo 1834
Mourne Anthony Crott Kilbride Mayo 1834
Mourne Anthy Crott Kilbride Mayo 1834
Mourne Daniel Rathoonagh Kilbride Mayo 1833
Mourne Anthony Crott Kilbride Mayo 1833
Mourne Daniel Rahoonagh Kilbride Mayo 1833
Mourne Anthony Crott Kilbride Mayo 1833
Mourne Anth Crott Kilbride Mayo 1833
Mourne Arthur Killerney Roscommon Roscommon 1824
Mourne Arthur Carramore Roscommon Roscommon 1824
thats as mush as i can offer, the more open you are eg i would check Genes Reunited if you can give a person and approx birth