I need an explanation. Please, excuse me if I am boring you on the Forum. 
to take leave = to leave it alone = to let it slip by = to forget about it
I suspect that this use is somewhat archaic.
The character wants to settle (deal with right now) the money situation instead of putting it off for later.
in regular usage, āto take leaveā means to go on vacation
Like what arw69733 said, it normally means to go on vacation.
Wrong! 
It means, in the lesson, that he left the room, the house, or the conversational setting; or wanted to do so, or wasnāt sure whether or not he could do so. You can see another example a few paragraphs down, by the way.
Now, in bureaucrat-speak, yes, taking leave means taking part or all of oneās annual vacation time.
LOL, did anyone read the content first?
āThe poor young man hesitated and procrastinated: it cost him such an effort to broach the subject of termsā¦ Yet he was unwilling to take leaveā¦ā
In the context of this quaint, archaic writing, ātaking leaveā has nothing to do with going on holiday/vacation. Neither was the āpoor young manā wanting to squeeze past the fat lady and exit the room, lol.
Rather, taking leave here means something like asking permission to interrupt the fat ladyās repetitive speech where she discussed all but what he needed her to say, which was in fact his amount of pay: āHe would have liked to hear the figure of his salary; but just as he was nervously about to sound that noteā¦ā
It must be my age (!) but I still remember my grandmother using that old-fashioned term.
Also, when law courts apply for leave to appeal, for example, the meaning of āleaveā is to get permission or right to appeal, rather than meaning getting time out to do so.
āTo take leaveā is used sometimes nowadays, but only as the go on vacation meaning. Creimann may be right in regards to the story though. I didnāt look too thoroughly myself. Whatever it means in the story, the thing to remember is that, people donāt use that exact expression in that exact way anymore.
You may be spot on there. The first instance of ātaking leaveā in the lesson might be about venturing to speak. There isnāt really much context to draw from, in the first paragraph. Later on, though, the same expression is used in relation to departing.
@ creimann - ah, now you get it. I concur, it was about venturing to speak. But also, one didnāt just interrupt back then - one would ask permission to butt in. Nowā¦if only my kids came on boardā¦
Nodding in agreement at the different second usage: the young man, that is, the young prospective student - not the same as the āpoor young manā who is in fact the new tutor, takes āleave of his motherā - where he departs from her presence. In other words, the boy leaves the room.
This is quite an advanced text for non-native English speakersā¦even English speakers wouldnāt necessarily have sophisticated nuanced understanding of this contentā¦LOL, being cheeky here.
Ah, itās very likely too in this context that ātaking leaveā meant the boy asked permission of his mother to leave prior to leaving the room, though itās unspoken. He would have been required by his mother to be there to meet the prospective tutor, and would have needed her assent to leave.
@Ninche - hereās my two centsā worth: being able to understand old-fashioned, out of date language has nothing to do with fluency, if thatās your primary goal.
I might one day speak fluent Japanese and Chinese, but not necessarily be able to read ancient classics. Reading out of date English is a bit like reading the King James Bible: for example, āJesus saidā¦prevent the children from coming to Meā, even though the meaning clearly implied in that context is "let the children!
The Pupil content has even more difficult terms of phrase than the ātaking leaveā. But of course, weāre all free to study whatever we like. I just note that you said on your profile that your main goal is fluency. Keep up the good work!
Thank you all! Yes, my main goal is fluency, but I want to be able to read old books, stories, etcā¦ (the language of literature, new AND old, as well)