How can I add just the second elements in lists of pairs?

I like to do this kind of thing by writing a custom function to perform the the desired operation on two given pairs. I then map this function over the two lists with MapThread. In most cases, by using Mathematica's argument-pattern destructuring, the custom function can be expressed extremely simply. This is one of those cases.

The custom function is

f[{x_, y_}, {_, z_}] := {x, y + z}

Note that the righthand side is a literal expression of the desired result.

Now let's contrive some test data. I write the generator so that two lists of pairs can easily be given any length.

With[{n = 3}, {a, b} = Partition[Partition[Range[4 n], 2], n]];
a
b

{{1, 2}, {3, 4}, {5, 6}}
{{7, 8}, {9, 10}, {11, 12}}

Now we MapThread the function f over a and b.

MapThread[f, {a, b}]

{1, 10}, {3, 14}, {5, 18}}

lst1 = {{1, 2}, {2, 4}};

lst1a = lst1;
lst1a[[All, 2]] *= 2;
lst1a

{{1, 4}, {2, 8}}

lst2 = {{3, 5}, {10, 2}};

lst1b = lst1;
lst1b[[All, 2]] += lst2[[All, 2]];
lst1b

{{1, 7}, {2, 6}}

ClearAll[f1]
f1 = Module[{l = #}, l[[;; , 2]] += #2[[;; , 2]]; l] &;

f1[lst1, lst1]

{{1, 4}, {2, 8}}

f1[lst1, lst2]

{{1, 7}, {2, 6}}

f1[a, b] (* a and b from m_goldberg's answer *)

{{1, 10}, {3, 14}, {5, 18}}

Also

ClearAll[f2]
f2 = Module[{l2 = #2}, l2[[All, 1]] = 0; # + l2] &;

{f1[lst1, lst1] == f2[lst1, lst1], f1[lst1, lst2] == f2[lst1, lst2], f1[a, b] == f2[a, b]}

{True, True, True}

Try this:

{{1, 2}, {2, 4}} + ({{1, 2}, {2, 4}} /. {x_,y _} -> {0, y})

(* {{2, 4}, {4, 8}}  *)

or this

(# + (# /. {x_, y_} -> {0, y})) &[{{1, 2}, {2, 4}}]

(* {{1, 2}, {4, 8}}  *)

Have fun!