The specific word choice depends on context, but the most important thing to remember is that your entire sentence must be in the same tense. If it's still the morning in question, you can use:
I have had so many interruptions this morning that I have scarcely done any work.
Later in the day you could still use the above, but you might instead say:
I had so many interruptions this morning that I did scarcely any work.
And, if you were referring to events that took place after the interruptions or that might have taken place if you had remained uninterrupted, on a subsequent day you could say:
I had had so many interruptions that morning that I had scarcely done any work.
You may notice that I modified the word order a bit from your original question. This is a matter of personal style; you can insert the adverb 'scarcely' in any of a few positions in the sentence without sacrificing much clarity, but some readers may find my order a bit more natural.