litre

For the word on its own? Absolutely. Even in the bible we find instances of variations of להחזיק used in the meaning "to strengthen" (as seen in Nehemiah 3, for example). In this specific instance, however, it's used as part of a well known phrase throughout the Talmudic and the later rabbinical literature (whose source is from Ezekiel, as noam_k helpfully noted).

I just did a search (using https://tablet.otzar.org, as I don't currently have access to my academic resources), and I couldn't find an instance where this specific phrase (להחזיק יד, and its variants. My search query was `&??חזיק{0}יד`, without the backticks) is used in the meaning "to strengthen the hand". All I could find was of the "to support" meaning. Can you point to specific references where this isn't the case?

Can you elaborate which "last point" you're referring to? I'm not sure how the Ezekiel reference contradicts anything I wrote.

The prayer book in this article is most likely a book with liturgical prayers. The text that the article discusses was just written on a leaf in that book (likely one of the opening or ending ones). It's quite common to find in books of the era notes about personal or communal events written on what were blank (or mostly blank) pages.

Not necessarily. The relevant part in the text reads

>> [...] טעונין בר ולחם ומזון ויין להחזיק יד עני ואביון ולכל איש אשר יחסר לו [...]

לְהַחֲזִיק literally means "to hold", though in some contexts (especially in older texts like this) it also means "to maintain" or "to support". While it's close to לְחַזֵּק, which means "to strengthen", להחזיק isn't usually used in that meaning (apart from lyrical contexts, perhaps).