I am interested in the solution of the following Fourier transformation
$$\int \frac{1}{|x|^{d+a}}e^{-ixk}d^dx ,$$
considering a general $d$-dimensional system with $a\in\mathbb{R}_+$.
How is this distribution called (to better search for ways to solve it by myself)?
Is there a textbook solution to this problem?
Is there already something on StackExchange regarding this issue?
On
Fourier transform: 
A case analysis is necessary. There is no general solution to this.
The first step is to consider the absolute value or normalization of the vector.
$d==-a$: the Fourier transform is then the Dirac delta function.
$d=-a+1, d=1$: $-\sqrt{\frac{2}{π}} (EulerGamma + \log[|k|)$
$d=a+2, d=1$: $-\sqrt{\frac{2}{π}} |k|$
and so on.
The notation is not so far distributed among the modern community of mathematicians. Thats is due to the special names such norms have. Such a notations saves confusion:
$Norm[{x, y, z}, p]$
Vectors are in most CAS treated as list and in programming languages too.
So I read the question as $d$ is the dimension of the vector space and $a$ is my $p$. Some authors use the term metrics is the norm has certain attributes, positive definite, triangle inequality and alike. $a=1$ is called absolute norm, $a=2$ is the euclidean norm and so on. $a=\infty$ is called the maximum norm most often of the absolute values.
Multidimensional Fourier transform:
The easiest once are the once for the circle and sphere and their surfaces.
Physicists call this function the inverse distance law. It appears in points or circle or sphere like problems. $a=2$ for physicists.
Fourier transform of $\frac{1}{\sqrt{x^2+y^2}}\rightarrow\frac{1}{\sqrt{v^2+w^2}}$
The picture shows the start of the long series of Fourier transform for $a=2$ for $d$ positive integer.
$$ FourierTransform[1, {x, y}, {u, v}]= 2 π DiracDelta[u] DiracDelta[v]$$
$$\frac{1}{\sqrt{2\pi}^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} 1 e^{i(v t_x+w t_y)}dt_xdt_y =2\pi\delta(v)\delta(w)$$
Symbolic Fourier transform are a domain of formulas collection of symbolic CAS, like Mathematica, Maple, Matlab and others.
For books have a look at:
Google books search for symbolic fourier transform. Nice is Abramovich & Stegun Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical ..., Springer Handbuch der Mathematik
To deal with higher d and real a is far beyond scopes for usual maths. For the RealAbs, absolute values on the reals, this already has numerical solutions.
Similar questions have indeed been asked here now and then, and part of the problem is indeed about keywords to refer to the question.
As commented, some relevant keywords are "homogeneous distribution", also "spherically/rotationally invariant distribution", and such.
An immediate technical obstruction to literal mathematical computation is that, even in the range $\Re(s)<n$, for local integrability, $1/|x|^s$ is not $L^1$ on $\mathbb R^n$. So its Fourier transform cannot be computed by the usual integral. It is also certainly not in $L^2$, so the Plancherel extension of Fourier transform also cannot quite succeed.
Nevertheless, with $\Re(s)<n$ for sure, integration-against $1/|x|^s$ gives a tempered distribution... so it has some kind of Fourier transform, albeit not easily computed by integration.
The idea of a proof is that Fourier transforms of rotationally invariant (tempered) distributions must be rotationally invariant, and homogeneity of degree $s$ (with $1/|x|^s$ as model) is mapped to homogeneity of degree $n-s$ (with $1/|x|^{n-s}$ as model). These two things can be proven in various ways...
Also, it might be good to have in hand a uniqueness result for homogeneous, spherically-symmetric distributions...
Then we "know" that for typical $s$ the FT of (finite part of) $1/|x|^s$ is some multiple (depending on $s,n$) of the (finite part of) $1/|x|^{n-s}$. The anomalous points where this meromorphic family of tempered distribution has poles (in $s$) produces residues which are multiples of $\Delta^\ell \delta$, and Fourier transforms that are integer powers of polynomials $|r|^2$.
The constants can be determined, in terms of the Gamma function, by integrating against Gaussians, since we explicitly know their Fourier transforms (depending on one's choice of normalization of everything...)