I was playing with some friends the other day and since we all listen to jazz fusion (like Herbie Hancock and Snarky Puppy) we were trying to play jazz standards in that style. However, none of us knew how to actually make music that sounds like funk-inspired jazz fusion. How do we make it sound like we're playing fusion?
Fusion in music means a musical style which is inspired from two or more different musical styles. I think you are trying to play music which have both funk and jazz influences.
The main ingredient of funk is rhythm. Try to incorporate funk rhythmic ideas into jazz standards.
Take a funk song and reharmonize with ideas inspired from jazz.
There can be different ways to approach fusion, but one thing will always be there: creativity. Listen to as many musical styles you like and understand them technically and spiritually. Hopefully you will find a way yourself.
For context, my teacher is Lenny White, most famous for being the drummer in legendary 'fusion' group Return to Forever. If you don't know him, look him up - he is a founding father of the music, and was present when Miles created this 'fusion' sound. I'm a saxophonist.
I'm 99% sure you guys don't have the rhythmic aspect. Getting your rhythm together is the first and most necessary step in recreating this music, before any harmony ("jazzy reharms", extensions, etc.) or any timbre concerns (what synth to use, how many pedals, etc). Practically, you should learn to play over a simple form as a tight rhythm section. Doing this requires all members, but especially your drummer, to study (listen, transcribe, and play along to) the history of funk drumming outside of jazz, and then do the same for Lenny et al. Your question was too general for me to give any other specific advice on that.
Regarding timbre and harmony, it is telling that Lenny dislikes having this music called fusion. as it is too general. Instead, he refers to it as 'jazz-rock'. The harmonic and melodic language for forms and improvisation was entirely established in the 60's by all the jazz musicians in that era. The timbre of the music was entirely established by rock and pop (funk, disco) musicians of that era. So again, study 60's jazz (Miles 2nd great quintet, Coltrane, et al.), and 70's funk and rock (Hendrix was a huge influence, James Brown et al). Then revisit your favorite jazz-rock albums.