#JavaScript (ES6), 33 bytes

Saved 1 byte thanks to @ShieruAsakoto

I/O format: array of words.

a=>a.map(g=w=>w&&w+g(w.slice(1)))

Try it online!

#JavaScript (ES6), 35 bytes

I/O format: array of words.

a=>a.map(w=>w.replace(/./g,"$&$'"))

Try it online!

JavaScript (ES6), 33 bytes

Saved 1 byte thanks to @ShieruAsakoto

I/O format: array of words.

a=>a.map(g=w=>w&&w+g(w.slice(1)))

Try it online!

JavaScript (ES6), 35 bytes

I/O format: array of words.

a=>a.map(w=>w.replace(/./g,"$&$'"))

Try it online!

#JavaScript (ES6), 45 bytes

s=>s.replace(/\S+/g,g=s=>s?s+g(s.slice(1)):s)

Try it online!

#JavaScript (ES6), 46 bytes

s=>s.replace(/\S+/g,s=>s.replace(/./g,"$&$'"))

Try it online!

#JavaScript (ES6), 33 bytes

Saved 1 byte thanks to @ShieruAsakoto

I/O format: array of words.

a=>a.map(g=w=>w&&w+g(w.slice(1)))

Try it online!

#JavaScript (ES6), 35 bytes

I/O format: array of words.

a=>a.map(w=>w.replace(/./g,"$&$'"))

Try it online!