Is it permissible to apply for additional financial allowances to cover monthly obligations, given the insufficiency of the current stipend alongside the rising cost of living?

If the intention is to request money from the state, then there is no harm in that, for you have a right to the state treasury. Requesting money from the ruler is permissible, as indicated by the Hadith (definition): "Begging is scratches by which a man wounds his face. So, whoever wishes may preserve his face, and whoever wishes may leave it. Except for a man who asks from a ruler, or in a matter from which he cannot escape." This is evidence for the permissibility of asking the ruler for Zakat (definition), or the Khums, or from the public treasury (Bayt al-Mal), as well as in cases of necessity and need from which there is no escape but to ask.