The text is available here. I'm not sure exactly which sentence is giving you trouble. I'll explain both of them, then:
At odd moments, Heinrich gently reminds her of that datum.
This means that Hannah's husband Heinrich reminds her of the fact that she is only human.
He points out a tic in her German or a run in her stocking.
The author gives us two examples of Hannah's imperfections that her husband points out to her:
(a) oddities or repetitions in her German
(b) a run in her stocking
I'll explain (b) first. Women used to have to wear stockings every day. Stockings were like thin tights, or very long, above-the-knee, thin socks. The way they were knitted was that if a tiny hole developed in the knitted fabric, the knitted structure would unravel in a narrow line in both directions -- up and down -- emanating from the hole. If you caught it in time you could mend it and prevent it from spreading further. If it was a long run, the mending was time-consuming.
Knowing that the tiniest of snags could ruin a stocking, was another constraint that prevented women from functioning freely. A woman had to move around her world carefully. Otherwise she'd ruin every stocking she ever put on, and she'd have to buy a new pair of stockings for every day of the week, for 365 days of the year.
It would be hard to justify feeling resentment towards the husband who points out the flaw in the stocking if he points it out "gently."
Now I'll explain (a). After doing some reading about Hannah Arendt, it seems clear to me that German was definitely her primary language. "A tic in her German" means some imperfection in her German. Why would she have some imperfections in her German, if German was her primary language, and she was a well-read intellectual who did a lot of writing and publishing?
I have two ideas about this.
First, it seems Königsberg, where she grew up, was pretty heterogeneous, linguistically, with Yiddish and Polish being reasonably common secondary languages there. Also, Wikipedia says that there was a "low German" dialect spoken there. Even if Hannah was using High German in school (which I think would have been the case), the dialect commonly used there may have been a bit different from the standard High German, and she may have absorbed some of these peculiarities.
Second, German grammar is tricky even for Germans. Often in those days, in a romantic couple where both parties were heavy duty intellectuals, it was generally the case that the balance of power favored the male member of the couple. It's not hard to imagine that Hannah's husband might have occasionally caught some slip-up in Hannah's writing (or even in her speaking).