Now that makes more sense you never said when deleting a products.
Thats a whole separate script
In adm-product.php find this
$ipaddr = findenv("REMOTE_ADDR");
$date = time();
$what = "Deleted product: $filename ($ptitle)";
$what = addslashes($what);
$query = "REPLACE INTO {$Globals['pp_db_prefix']}admlog (adminuser,ip,datestamp,description) VALUES ('{$User['username']}','$ipaddr','$date','$what')";
$resultb = ppmysql_query($query,$link);
change to this
$username = addslashes($User['username']);
$ipaddr = findenv("REMOTE_ADDR");
$date = time();
$what = "Deleted product: $filename ($ptitle)";
$what = addslashes($what);
$query = "REPLACE INTO {$Globals['pp_db_prefix']}admlog (adminuser,ip,datestamp,description) VALUES ('$username','$ipaddr','$date','$what')";
$resultb = ppmysql_query($query,$link);