It's odd that $month starts 0-based and ends 1-based. You also assume $monthsback is a small number. Fix:

use POSIX qw( floor strftime );

my $monthsback = 3;   # 0=this month -1=last, -2, ...

my ($day,$mon,$year) = (localtime())[3,4,5];
$mon  -= $monthsback;
$year += floor($mon / 12);
$mon  = $mon % 12;

print(strftime("%Y-%m", 0,0,0, $day,$mon,$year), "\n");
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A more general solution:

use POSIX qw( floor strftime );

my $mon_delta = -3;  # ..., -2, -1=last, 0=this month +1=next, +2, ...

my ($day,$mon,$year) = (localtime())[3,4,5];
$mon  += $mon_delta;
$year += floor($mon / 12);
$mon  = $mon % 12;

print(strftime("%Y-%m", 0,0,0, $day,$mon,$year), "\n");
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