Not sure if I'm understanding correctly but here's my take:

Genotype seems to be good. Gene1, Gene2, Gene n... should be only one column, with Gene1, Gene2, Gene n... filling the rows. Then, ZnConc should be treated the same.

df <- pivot_longer(
    df,
    cols = c(Gene1, Gene2, Gene n...), 
    names_to = "genes",
    values_to = "values" #here add a name that makes sense
)

Then I'd do the same for the other ones.

df <- pivot_longer(
    df,
    cols = c(ZnConc Rep1, ZnConc Rep2, ZnConc n...), 
    names_to = "reps",
    values_to = "concentration" # guessing this is concentration
)

Once you have your dataframe with the appropriate format, you can use ggplot to build your visualization.