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The solution

array_sum(
    array_foreach( 
        string_to_array ( "list of polygons" , ','),
        attribute (get_feature (@layer, 'name', @element), 'area')
    )
)

Explanation

The error in your expression is that you use the funtion area() with the argument @element, but @element returns the name of a polygon, whereas area() expects a polygon's geometry as input.

The functions area() as well as $area calculate the area from the geometry. You, however, just want to add values that already exist in an attribute called area.

How to implement

1. So inside the array_foreach() function, you get the name of one of the polygons. You then must identify the feature (line in your attribute table) of this element - like line 2 for feature B in your example. Use get_feature() for that.

2. Than for this feature, get the value of the attribute named area, using the function attribute().

As you remarked in your comment, be aware where to use double quotes " and where to use single quotes '. Best see help for QGIS expression functions. A field name in single quotes returns the name of the field, a field name in double quotes returns the value stored in that field for the current feature.