  We introduce a system combining the quadratic self-attractive or composite
quadratic-cubic nonlinearity, acting in the combination with the fractional
diffraction, which is characterized by its L\'{e}vy index $\alpha $. The model
applies to a gas of quantum particles moving by L\'{e}vy flights, with the
quadratic term representing the Lee-Huang-Yang correction to the mean-field
interactions. A family of fundamental solitons is constructed in a numerical
form, while the dependence of its norm on the chemical potential characteristic
is obtained in an exact analytical form. The family of \textit{quasi-Townes
solitons}, appearing in the limit case of $\alpha =1/2$, is investigated by
means of a variational approximation. A nonlinear lattice, represented by
spatially periodical modulation of the quadratic term, is briefly addressed
too. The consideration of the interplay of competing quadratic (attractive) and
cubic (repulsive) terms with a lattice potential reveals families of single-,
double-, and triple-peak gap solitons (GSs) in two finite bandgaps. The
competing nonlinearity gives rise to alternating regions of stability and
instability of the GS, the stability intervals shrinking with the increase of
the number of peaks in the GS.
