言い訳が また上手くなってく 理由の
Is く there to connect the sentence? I'm probably overthinking this but it bugs me not understanding the purpose of this く.
By the way, this is a lyric from this song; https://www.youtube.com/watch?v=EejcsaMDduQ
Without listening to the song, this is simply changing うまい (上手い) into "adverbial" use to connect with なる (as all イ-"adjectives" do).
うまい → うまくなる
Oops, wrong く. This is just the contracted form of なっていく, which is modifying 理由.