Its wrong to assume Persian was polular language of the majority people during the moghul . It may be the official language of the courts as the being language of the turkish origins Mogul ruling class, it the local dialects that were used extensively as language of communication of the common citizens.
Let me give several examples.
Gita Gobinda wrote in praise of Lord Jananath in by Poet Jaydeva in 16 century Odisha was written in Sanskrit and old odiya language, then one can easily argue odisha came under muslim(mogul)rule in 1565 long after much of northwest/north india was already under them for several centuries.
Up north,
Tulsi Dash wrote "Ram charita manas" in local KharliBoli,not in Persian.
So are the Doha written by Sant Kabir.
Guru Grant Sahib of Sikh religion started in Punjab in the same period was written in Gurmukhi, not Persian.
I wont go down south cause the influence of Persian was even less there though they too came under moghul rule,and later on, the Nizams of Daccan.
The argument is though Persian or Arabic for that matter were the official languges of that era and probably had influence and same import what English language today has, though to very less extent,it was never the language of the common man,atleast for majoirty hindus of that time.