This translation is British English, I have never heard, “someone has toothache”. It would always be “someone has a toothache” in American English. It’s the same with hospital, in America we go to the hospital, in Britain they just go to hospital. So if you are American, you will learn some British English here along with Gaelic, no extra charge.
I agree, wouldn't an English speaker say "a toothache". I can understand the reasoning for keeping a translation like "Tha Màrtainn a faicinn...." as "Martin is seeing..." as opposed to ''sees'' but an article of some sort seems warranted. If I'm translating it for someone in English I'm saying Martin has a toothache.
Part of it is this speaker. He's pronouncing it very much like a -k sound, as if it were spelt with a G.
The other speakers make a clearer distinction with either a velar fricative or a voiced velar stop (which doesn't normally exist in Gaelic, so the distinction is retained)