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Other issues to worry about are VAT if you are VAT registered. As the supply of services is to a business customer, then the "place of supply" is where the business customer belongs, i.e. Ireland and outside UK VAT scope and no VAT is charged.
On conclusion there a number of ways of making your structure more tax efficient and I shall list them below:
Option 1: If staying less than 183 days in Irish tax year. It best to keep the way things and use your UK firm providing services to Ireland. You are unlikely to pay tax in Ireland.
Option 2: If you are staying longer, you may need consider using a smaller agency or umbrella companies or one ideally in offshore centres. OR
Option 3: If you are staying longer, you can set up your own Irish LTD. There is lower tax in Ireland. However there are things to watch like ensuring you comply with HMRC offshore regulations and anti-avoidance tax laws. In addition Irish companies have other specific requirements like >1 director etc.
Option 4: If you are staying longer or permanently working there, then there is no point in having a UK company rather maintain an Irish Company instead.
Option 5. You can use offshore schemes. Most people who work across jurisdictions find setting up a company in places like Channel Islands or low tax jurisdiction makes sense. The very fact that you are providing international services justifies that- however you also need to comply with all the countries you provide the service to ensure you are not outside their scope.
My final comment to consider the above, but before you make any decision on the options please hire an offshore tax adviser whom you will discuss in detail of your intentions and he will advise on the risk and rewards of each option above.
Hope this helps.