How to row sum with factors in R?

I have data like so:

df <- data.frame(id = c("001", "002", "003", "004"),                 banana = c("banana", NA, NA, NA),                 apple = c(NA, "apple", "apple", NA),                 orange = c("orange", NA, NA, NA),                 mango = c(NA, NA, NA, "mango")) id banana apple orange mango001 banana    NA orange    NA002     NA apple     NA    NA003     NA apple     NA    NA004     NA    NA     NA mango

I'd like to consolidate the variables into a variable flavor in the following manner:

 id banana apple orange mango flavor001 banana    NA orange    NA  mixed002     NA apple     NA    NA  apple003     NA apple     NA    NA  apple004     NA    NA     NA mango  mango

How do I do this? Any help is appreciated, although a generalisable dplyr solution would be great.

Edit: I need a solution robust to having other columns in my dataframe too.