Wheezing along the way, I keep running, gritting my teeth.

Literal

Wheeze-wheeze saying-[while-ながらも], determination [obj-を] put-in [and-て] keep-running.

ぜいぜい is a reduplicated mimetic for wheezing or labored breathing, typically used with the [mimetic]+いう construction for 'making the X sound.' ~ながらも '(though / even) while X' layers a mild concessive onto the ながら 'while' pattern. 気合を入れる is the set expression 'to summon one's will / psych oneself up,' literally 'insert fighting spirit.' 走り続ける is a compound 走る + 続ける 'keep running' — the productive ~続ける 'keep doing X' suffix.