EMGM's
map demands traversion function to be non-polymorphic, i.e. type-checker
fails with the message, complaining it cannot match `E a` against
`E Name`, against `E Salary` etc.

I'm wondering if you tried everywhere' (or everywhere) [1]. Here's one solution, but I'm not sure if it does what you what it to.

--

{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE TemplateHaskell #-}

module Rows where

import qualified Generics.EMGM as G
import Generics.EMGM.Derive

data Row = Row (Either (Maybe Int) (Maybe String)) (Either (Maybe Int) (Maybe Float)) (Either (Maybe Int) (Maybe Integer))
deriving Show

$(derive ''Row)

gmap :: (Rep (Everywhere' (Either (Maybe Int) (Maybe a))) Row) => (Either (Maybe Int) (Maybe a) -> Either (Maybe Int) (Maybe a)) -> Row -> Row
gmap = G.everywhere' -- top-down

readRow :: [String] -> Row -> Row
readRow l = gmap app
where
    app :: Either (Maybe Int) (Maybe String) -> Either (Maybe Int) (Maybe String)
    app (Left (Just ri)) = Right (l `atMay` ri >>= G.read)
    app x = x

atMay :: [a] -> Int -> Maybe a
atMay = undefined

--

This appears to implement your desired functionality. Here are some points to note about what I did to get it working:

* EMGM has problems resolving type synonyms, so I expanded your E here.
* I just defined gmap to show what the type signature would be here. You could get rid of gmap and just use everywhere'.
* I used everywhere' instead of everywhere, because you appear to want a top-down traversal. Depending on your types, it may not matter.
* I gave app a concrete type signature, because as you noted, EMGM needs to be non-polymorphic here.
* I also gave app a fallback case, so you don't get any unexpected surprises at runtime.
* I used EMGM's read function [2] which seemed to be what you wanted for readMay. You could still use readMay here, of course.

[1] http://hackage.haskell.org/packages/archive/emgm/0.3.1/doc/html/Generics-EMGM-Functions-Everywhere.html
[2] http://hackage.haskell.org/packages/archive/emgm/0.3.1/doc/html/Generics-EMGM-Functions-Read.html

Regards,
Sean