I think this exact query was posted before, and I think I replied to it then. Anyway:
1. If the solicitor hires you, he\she is liable to pay your agreed fee.
2. If the solicitor retains you on behalf of joe bloggs (as is more common), then joe bloggs is liable to pay your agreed fee.
3. A successful plaintiff (usually in a compensation claim) will be awarded their costs. Sometimes, they will be awarded less than their costs; for example, you might bill €1,000 and the Taxing Master might only award your customer €800 toward this. You are not obliged to accept the smaller sum - the costs awarded belong to the Plaintiff; the liability to pay you is the liability of the Plaintiff.
4. Many people who do a lot of court work understand the taxing of costs; In some cases, implicitly or explicitly, they agree that their fee will be reduced if the amount awarded does not cover it. However, I would consider it very bad form indeed for a solicitor to unilaterally impose a reduction on the amount remitted to you, as seems to be happening. Mind you, if you have tolerated this regularly, it might be that you have implicitly agreed to it. If you don't like this, get a proper letter of engagement template done and issue it at the start of each job for a solicitor so as to avoid future misunderstanding.
The important point is that you are under no obligation whatever to justify your fees to the Taxing Master. The person who hired you has to justify these fees if they want to recover the money from the other side. The issue of whether they justify the fees or not does not alter their liability to you to pay whatever fee they have explicitly or implicitly agreed to pay.