You don't need to read the MAP file for every line of the input, just read it once

#!/usr/bin/perl
use strict;

open MAPFile,'<',$ARGV[2] or die "$ARGV[2] : $!";
my %hash  = (); # Cost Center, Activity
my %hash1 = (); # Account
while ( <MAPFile> ) {        
  chomp;
  my ($CC,$Act,$NEWCC,$NEWAct,$Acct,$NEWAcct) = split(',',$_);
  $hash{$CC}{$Act}[0] = $NEWCC;
  $hash{$CC}{$Act}[1] = $NEWAct;
  $hash1{$Acct}       = $NEWAcct;
}
close MAPFile;
  
open SourceFile, '<', $ARGV[0] or die "$ARGV[0] : $!";
open TargetFile, '>', $ARGV[1] or die "$ARGV[1] : $!";

my $count = 0; my $changed = 0; my $changed1 = 0;
while (<SourceFile>) {
  ++$count;
  my ($Year,$HSP_rates,$Curr,$Scenario,$Version,$Product,
   $CostCenter,$Activity,$Entity,$Acct,$BegBal,@mth) = split(',',$_);
   
  my ($NewCC,$NewAct,$NewAcct); 
  if ( exists $hash{$CostCenter}{$Activity} ){
    $NewCC   = $hash{$CostCenter}{$Activity}[0];
    $NewAct  = $hash{$CostCenter}{$Activity}[1];
    print "line $count : updated $CostCenter to $NewCC\n";
    ++$changed;
  } else {
    $NewCC   = $CostCenter;
    $NewAct  = $Activity;
  }
  
  if ( exists $hash1{$Acct} ){
    $NewAcct  = $hash1{$Acct};
    print "line $count : updated $Acct to $NewAcct\n";
     ++$changed1;
  } else {
    $NewAcct  = $Acct;
  }
  
  print TargetFile join ',',
   $Year,$HSP_rates,$Curr,$Scenario,$Version,$Product,
   $NewCC,$NewAct,$Entity,$NewAcct,$BegBal,@mth;   
}
print "$count lines processed 
 $changed  changes to CostCenter,Activity codes 
 $changed1 changes to Account codes\n";

close SourceFile;
close TargetFile;
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Update ; %hash1 added for Acct mappings