More generally seen, your problem is, that you want to predict a multi-variate / multi-dimensioan lable, but your algorithm only supports uni-variate / 1-dimensional target variables.
I see two direct approaches that you could try:
You could train $H$ independent XGBoost Models, one for each target dimension.
sklearn already provides a wrapper for this:
model = MultiOutputRegressor(XGBRegressor())
Flatten the output
You could turn each sample into $H$ different samples, one for each output dimension. In order to distinguish these dimensions, you could add the index of the output as another feature.
In other word, your $(N,T)$ input would be transformed into an $(N\cdot H, T+1)$ input and your target into an $(N\cdot H)$ vector.
For example, the data
X = [[1, 5, -3],
[2, 4, 6]]
y = [[4, 6],
would be transformed into
X = [[1, 5, -3, 1],
[1, 5, -3, 2],
[2, 4, 6, 1],
[2, 4, 6, 2]]
y = [4, 6,13,20]