He took an active part in the following year in the campaign against Gerald FitzGerald, 15th Earl of Desmond; but during the absence of Sir Nicholas Malby, Lord President of Connaught, in the winter of 1580–81, he acted as governor of that province, and pursued the Burkes and other disturbers of the peace.
In 1581–82, he was occupied, apparently between Clonmel and Kilmallock, in watching the movements of the Earl of Desmond, and on the retirement, of Captain John Zouche in August 1582, on account of ill-health, he became colonel of the forces in Munster.
In consequence of the appointment of Thomas Butler, 10th Earl of Ormond as governor of Munster, Norris was able, early in 1583, to pay a brief visit to England.
In the autumn of 1584 he took part in Sir John Perrot's expedition against the Scots in Antrim, and in scouring the woods of Glenconkeyne in search of Sorley Boy MacDonnell he was wounded in the knee with an arrow.
Under instructions from England, Norris, in March 1587, arrested James Fitzedmund Fitzgerald, seneschal of Imokilly, Patrick Condon, and others, whose loyalty was at least doubtful.
The marriage of Ellen, daughter and sole heiress of Donald McCarthy, 1st Earl of Clancare, was a politically sensitive topic and Norris himself rejected her as his bride.
The Spanish Armada was over but the air was still full of rumours of invasion, and in 1589–90 Norris was engaged with Edmund Yorke, an engineer who had been sent over from England expressly for the purpose, in strengthening the fortifications of Limerick, Waterford, and Duncannon.
In the following year he served under his brother, Sir John Norris, against the Earl of Tyrone, and was wounded in the thigh in the engagement that took place halfway between Newry and Armagh on 4 September.
He assisted Sir John Norris as commissioner for the pacification of Connaught in June 1596, but, in August, he was engaged in repelling an incursion of the MacSheehys and O'Briens into Munster.
Norris is mentioned by their mutual friend Lodowick Bryskett as one of the company to whom Edmund Spenser on a well-known occasion in the late 1580s unfolded his project of writing The Faerie Queene.