The highlighted word is not relevant to understanding the response from Rabbi Hoffman as can be seen from the content of his answer.
He understands the question to be dealing only with the subject of whether a person threatening to kill themselves is considered a circumstance of danger to life that would warrant someone else to violate Shabbat. In the example from the question, the method of violation would be writing on Shabbat. Where that would take place is irrelevant from what appears in his answer.
That said, it seems unlikely that a religiously observant Jewish family from late 19th century Germany would be sending their young son to attend a trade school that would require him to violate the Shabbat. And it seems even less likely that his father would be advocating for him to violate Shabbat.
Additionally, trade/business school would be spelled with Yiddish/Hebrew letters האנדלשול or האנדלשולע. Spelled in German Handelsschule. But this run-on typo is spelled בהאנרעלזשולע.
Since the remainder of the question and answer are in Hebrew and only this phrase is in Yiddish, it suggests that the Yiddish phrase is clarifying how the father was telling his young son to write on Shabbat. The Yiddish ending of the phrase is זשולע, pronounced zhule, not shule. זשולע or זשעלע means jelly.
It looks like it is a run on of the Yiddish phrase בהאנד על זשולע or בהאנט על זשולע.
The more reasonable scenario is that the father is telling his young son to write with his hand in jelly (שרייבן מיט א האנד על זשולע) on Shabbat. Perhaps he was learning the laws of Shabbat with him over the Shabbat meal. The child refused because he thinks it is a violation of Shabbat and a quarrel ensues.
This actually relates to the concept of what constitutes writing and erasing in connection with food on Shabbat like is discussed in Mishnah Berurah 340:15. As is explained there, writing is defined as letters or pictures written with a separate substance, such as frosting on a cake. Shapes etched into a food (or if the food itself is made into a particular shape) are not considered “writing” and so may be destroyed.
The mother is objecting to her young son refusing to obey his father and says that if the child doesn't obey, she will kill herself.
The question revolves around whether the child should potentially violate Shabbat in order to save the life of his mother when she threatens suicide.