I don't want people to think I can't do my job, so I'll work hard.
I don't want people to think I'm incompetent, so I'll do my best.
ーI don't want people to think I can't do my job, so I'll work hard.
「仕事ができないと思われたくないから、頑張る」
can't do one's job で「仕事ができない」
to work hard で「一生懸命働く」=「頑張る」
ーI don't want people to think I'm incompetent, so I'll do my best.
「無能だと思われたくないから頑張る」
incompetent で「無能な・役に立たない」
to do one's best で「精一杯頑張る」
ご参考まで!