What does 등글개첩 mean?

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ANSWER: a young mistress (concubine) of an old man

But what is the literal meaning of 등글개첩?

Well, 등 means back, and 긁다 means to scratch, so the 등글 in 등글개첩 literally means back scratching, but I think there are three possibilities for the meaning of 개첩.

First, 개 is a pure Korean suffix that means machine or instrument, as in the Korean word for wing, which literally means a flying (날) instrument (개). That means 등글개 could be translated as a back (등) scratching (글) instrument / machine (개). And since 첩(妾) means concubine, that would mean 등글개첩 would be translated as a concubine who is a back-scratching machine. The suggestion is that a young concubine would have few other duties but to scratch an old man's back since old men typically have lost much of their desire for sex.

Second, 애첩(愛妾) is a Sino-Korean word that means one's (favorite) concubine. It literally means a loved (愛) concubine (妾). When the 긁 in 긁다 is placed in front of 애첩, the combination would be pronounced /글개첩/ since the ㄱ in the bottom half of 긁 would attach to the 애 in 애첩 to form 개. Therefore, 등글개첩 could be translated as a favorite back-scratching concubine.

Third, since 등글개첩 means a YOUNG concubine of an old man, it seems possible that 애첩 could mean young concubine since 애 is a pure Korean suffix that means young, as in 애호박, which means a young (애) pumpkin (호박), or 애벌레, which means a young (애) insect (벌레) or larva. As was the case with the 애 in 애첩(愛妾),  favorite concubine, the ㄱ in 긁 would also attach to the 애 meaning young to form 개; therefore, 등글개첩 would translate as a young back-scratching concubine.

Of the three above suggestions for the literal meaning of 등글개첩, I personally like the first one: a concubine who is a back-scratching machine.