Is it permissible to take a portion of the money requested from others to help another person, knowing that the intention was to help that person fully, but an urgent need arose to take a portion of the money while ensuring that the beneficiary receives their full due?

What you asked about has three cases:

First: If the money is a gift for you, then you may dispose of it as you wish, and you may give it to the needy.

Second: If the money is for you to spend on the needy in general, and no one specific was designated for you, then you may spend it on your personal needs if you are truly in need of it to a religiously significant extent (basic necessities of life).

Third: If the money is specifically for the mentioned needy person, then you are an agent on their behalf, and it is not permissible for you to spend it on your personal needs. If you violate this and spend it on your own behalf or for another person, then you are sinful, and you must repent to Allah and immediately deliver the money to the one it was sent for.