Adding seek doesn't solve it.

#!/usr/local/bin/perl

$logFl = "xyz.log";
open (FIRSTFH, "> $logFl") || die "couldn't open file...$!\n";
print FIRSTFH "This is from first script\n";
system("script2.pl $logFl");
seek (FIRSTFH, 0, SEEK_END);
print FIRSTFH "This is second print in first script\n";
~
[download]

Now output is
$ cat xyz.log
This is second print in first script
m second script

Note that some part of print message in second script is not printed