明後日こづかいの一万円を自由に遣わしてくれるように母にいった。

。

I told my mother she should let me use the 10,000-yen allowance freely the day after tomorrow.

Literal

Day-after-tomorrow allowance [possessive-の] 10,000-yen [object-を] freely let-use [so-that-ように] mother [dative-に] said.

遣わす (つかわす) is an old/somewhat literary causative of 使う ('to use') — 'to let use / to dispatch.' ~ように言う is the indirect command pattern. Combined, it means 'told mother to let me use...' All in all, a slightly antiquated phrasing — modern casual Japanese would use 使わせてくれる or similar. こづかい ('pocket money, allowance') is a vocabulary word most learners will encounter early.