use strict;
use warnings;
use Data::Dump qw/pp dd/;


# --- input data

my @sorted_data =
  (
   {
    'count_payment' => '',
    'count_banking' => '31 (62.00)',
    'name' => 'Parking Eye [LTD] One',
   },
   {
    'count_payment' => '10 (144.00)',
    'count_banking' => '',
    'name' => 'Parking Eye [LTD] Two',
   },
   {
    'count_payment' => '2 (80.42)',
    'count_banking' => '',
    'name' => 'Parking Eye [LTD] Three',
   },
   {
    'count_payment' => '',
    'count_banking' => '4 (982.00)',
    'name' => 'Parking Eye [LTD] Two',
   }
  );



# --- join elements into temporary hash by name

my %tmp_hash;
my $order = 0;

for my $record (@sorted_data) {
    my $name = $record->{name};

    # create new hash only if $name unknown
    $tmp_hash{$name} //=
      {
       name   => $name,
       _order => $order++,              # preserve original order
      };

    # concat other elements
    for (qw/count_banking count_payment/) {
        $tmp_hash{$name}{$_} .= $record->{$_};
    }
}


# --- copy hash values by order

my @new_sorted_data =
  sort { $a->{_order} <=> $b->{_order} }
  values %tmp_hash;


# --- discard temporary _order element

delete $_->{_order} for @new_sorted_data;

# --- dump result

pp \@new_sorted_data;
[download]

Here a cleaner and more generalized version, also closer to Schwartzian transform

use strict;
use warnings;
use Data::Dump qw/pp dd/;


# --- input data

my @sorted_data =
  (
   {
    'count_payment' => '',
    'count_banking' => '31 (62.00)',
    'name' => 'Parking Eye [LTD] One',
   },
   {
    'count_payment' => '10 (144.00)',
    'count_banking' => '',
    'name' => 'Parking Eye [LTD] Two',
   },
   {
    'count_payment' => '2 (80.42)',
    'count_banking' => '',
    'name' => 'Parking Eye [LTD] Three',
   },
   {
    'count_payment' => '',
    'count_banking' => '4 (982.00)',
    'name' => 'Parking Eye [LTD] Two',
   }
  );



# --- join elements into temporary hash by name
my %by_name;
my $order = 0;

for my $record (@sorted_data) {

    my $name = $record->{name};

    # create new hash only if $name unknown
    $by_name{$name} //=
      {
       hsh => { name   => $name },
       ord => $order++,               # preserve original order
      };

    # concat other elements
    for my $field ( keys %$record ) {
        next if $field eq 'name';
        $by_name{$name}{hsh}{$field} .= $record->{$field};
    }

}


# --- copy hash values by order (Schwartzian transform)

my @new_sorted_data =
  map  { $_->{hsh} }
  sort { $a->{ord} <=> $b->{ord} }
  values %by_name;


# --- dump result
pp \@new_sorted_data;
[download]

Cheers Rolf
_{(addicted to the Perl Programming Language :)
Wikisyntax for the Monastery}

Thank you so much Rolf. Thank you so much for the great support provided.