In addition to the reference provided by @MikeHoneychurch, your question about
seeking $f(s)$ given the moments or projections or linear combinations $g(t)$,
falls into the same category as
question about multi-peak fitting. There, the $f(s)$ are Gaussian-like shapes.
Are you working with discrete samplings of the data $g(t)$? or are you looking
for theoretical, continuous forms of $g(t)$?
If discrete, then the integral equation may be transformed to a matrix system
by partitioning the $s$-axis into discrete intervals. The function $f(s)$ may be
represented on the partitioned $s$-axis as a comb of delta-functions, or narrow
rectangular shapes, or other form. Finding the integrals with the known kernel
functions $K(t,s)$ generates a matrix system $A\cdot x=b$. Mathematica has many built-in
functions for solving discrete system, with and without constraints.
I have been working with such equations for many years, especially for systems
where $x$ is non-negative. The book, Solving Least Squares Problems, and
fortran code by Lawson and Hanson may be useful. Michael Woodhams implemented a key routine, NNLS, in Mathematica
The paper by Istratov and Vyvenko (Exponential analysis in physical phenomena.
Rev. Sci. Instrum. 70(2), 1233 (1999)) is a good review of techniques.