Von Neumann Neighborhood

APL (Dyalog Extended), 23 bytes

-4 thanks to ovs.

Anonymous tacit infix function, taking coordinates of a cell and radius as left and right arguments. Requires 0-based indexing.

{⍺∘+¨⍵-⍸⍵≥+/¨|∘.,⍨⍵…-⍵}

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{…} "dfn"; left argument is ⍺ and right argument is ⍵:

 ⍵…-⍵ inclusive integer range from radius to negative radius

 ∘.,⍨ Cartesian selfie product

 ⊢m← assign to m and pass that

 | absolute values

 +/¨ sum each (gives matrix of Manhattan distances)

 ⍵≥ indicate which ones are less than or equal to the radius

 ⍸ ɩndices where true

 ⍵- subtract from radius

 ⍺∘+¨ add the cell coordinates to each

05AB1E, 14 13 bytes

-1 thanks to ovs

D(ŸãʒÄO¹s@}€+

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Takes input as range, coordinates

Explanation:

D duplicate the range in the stack
( negate
Ÿ push the range [-range..range]
ã Cartesian power
 ʒ filter
  Ä absolute value, vectorizes over each of the coordinates
   O sum - distance from 0,0
    ¹s@ less than or equals to the range
 } end filter
€ map
+ add, with the implicit center coordinate

Perl 5, 64 bytes

sub n($x,$y,$d){map{//;map[$x+$',$y+$_],-$d+abs..$d-abs}-$d..$d}

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Saved 4 bytes, thanks to Xcali.