Modify each element depending on previous elements

L = {{h -> 1}, {h -> 2}};

Y = Thread[y -> Flatten[Accumulate[Values@L ] - Values@L/2]];

Join[L, List /@ Y, 2]

 {{h -> 1, y -> 1/2}, {h -> 2, y -> 2}}

Or

MapThread[Append] @ {L, Y}

 {{h -> 1, y -> 1/2}, {h -> 2, y -> 2}}

Or

Flatten /@ Thread @ {L, Y}

 {{h -> 1, y -> 1/2}, {h -> 2, y -> 2}}

Clear[h, y]
a = {1, 2};
b = Accumulate[a] - a/2;
Transpose[{Thread[h -> a], Thread[y -> b]}]

{{h -> 1, y -> 1/2}, {h -> 2, y -> 2}}