I tried asking the assembly member to help me find a job, but was brushed off dismissively.
Literal
Assembly-member [to-に] employment [genitive-の] help [object-を] tried-asking [but-が] nose [with-で] was-brushed-off.
鼻であしらう (はなであしらう, 'to brush off with one's nose' = 'to snub / to dismiss contemptuously') is a vivid idiom. あしらう means 'to deal with / to handle (someone),' and 鼻で adds the contemptuous manner — treating someone as beneath notice. 就職口の世話 (しゅうしょくぐちのせわ, 'help finding a job opening') is a compound: 就職口 (job opening) + 世話 (assistance). 頼んでみた ('tried asking') uses ~てみる (try doing). The passive-like われた (= あしらわれた) marks the speaker as the recipient of the snub.