The variable used as a handle to an open file is . . . . . . . .

$dbh: This variable typically represents a database handle in Perl when working with database connections using Perl's DBI (Database Interface) module. It is not used as a handle for file operations.
$sth: This variable typically represents a statement handle in Perl when executing SQL statements using Perl's DBI module. It is used to handle the result set of a query, not for file operations.
$fh: In Perl, the convention for a file handle variable is often $fh, where fh stands for "file handle." It is used to represent an open file for performing file operations such as reading from or writing to the file.
$h: This option is incorrect. While $h is a generic variable name, it is not commonly used specifically as a file handle in the context of file operations in Perl or MySQL.

Therefore, the variable used as a handle to an open file is Option C: $fh.