It would depend on the nature of the work you do to earn the “UK” income, as to where it’s taxable and whether or not it has to be registered as a business activity in France. Eg if it is an online business activity that you carry out from France on a self employed basis, the activity would have to be registered in France and you would pay both tax and cotisations on it here, It would be nothing to do with the UK; the fact that your clients may be in the UK, makes no difference. However even that is not straightforward, because in France one individual cannot register 2 separate entreprises individuelles, so if you already have a siret number for your EIRL you couldn’t register another completely different self-employed business.
If you are working from France as an employee of your own company or someone else’s company you would need to be declared to URSSAF as an employee and pay cotisations that way, also income tax.
If the work is actually carried out in the UK, say if you go back to the UK and work from an office there 3 or 4 days a week, then the income would be taxable in the UK (but if you’re tax resident in France, would still declared as income in France as already taxed in the UK). The complication here would be healthcare. While the UK is a member of the EU and EU regulations apply, you cannot simultaneously belong to the social security system of 2 different states. So an agreement would have to be reached between the French and UK authorities as to where you are primarily resident, which would depend on where your family lives/where you spend your time/where you earn most of your income. And the rest would follow on from that (the state where you’re deemed to be resident and/or carrying out most of your work would be responsible for your social security, and that state would the issue you with an A1/S1 document to cover you for the periods when you work in the other state).
Trying to keep a foot each side the Channel is always complicated and the devil is in the detail, it can work out advantageous but you can also come badly unstuck. If you really want to do this, you need to get proper advice from this from an advisor, and provide him with exact details of where you’ll be spending your time and what your employment status will be. There’s also the question of CSG charged in France on UK income if you belong to the French healthcare system.