You can think about this in two ways:
Braket Notation
Quantum operators are linear, meaning any operator $\hatΩ$ obeys the following relation:
$$\hatΩ \left[ α|ψ⟩ + β|φ⟩ \right] = α \left(\hatΩ|ψ⟩\right) + β \left(\hatΩ|φ⟩\right) $$
In other words, you can just pull the constants out in front, calculate out the results of $H|0⟩$ and $H|1⟩$ as usual, then re-distribute your constants.
Generally $α$ and $β$ can be any complex number at all, but in your case $α=\frac{1}{\sqrt{2}}$ and $β=\left(\sqrt\frac{2}{7} + \frac{i}{\sqrt{7}}\right)$.
Matrix Notation
Matrix operations are designed to make these linear operations easier to think about. The vector you are trying to apply $H$ to can be written as:
$$ |Ψ⟩ = \left[ \begin{array}{c} α \\ β \end{array} \right] $$
Meanwhile, $H$ can be written as:
$$ H = \frac{1}{\sqrt{2}} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] $$
The final result is:
$$ H|Ψ⟩ = \frac{1}{\sqrt{2}} \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{c} α \\ β \end{array} \right] = \frac{1}{\sqrt{2}} \left[ \begin{array}{c} α+β \\ α-β \end{array} \right]$$
So, the final answer is:
$$ H|Ψ⟩ = \frac{1}{\sqrt{2}} \left[ (α+β) |0⟩ + (α-β) |1⟩ \right] $$