  We build a systematic calculational method for the covariant expansion of the
two-point heat kernel $\hat K(\tau|x,x')$ for generic minimal and non-minimal
differential operators of any order. This is the expansion in powers of
dimensional background field objects -- the coefficients of the operator and
the corresponding spacetime and vector bundle curvatures, suitable in
renormalization and effective field theory applications. For minimal operators
whose principal symbol is given by an arbitrary power of the covariant
Laplacian $(-\Box)^M$, $M>1$, this result generalizes the well-known
Schwinger--DeWitt (or Seeley--Gilkey) expansion to the infinite series of
positive and negative fractional powers of the proper time $\tau^{1/M}$,
weighted by the generalized exponential functions of the dimensionless argument
$-\sigma(x,x')/2\tau^{1/M}$ depending on the Synge world function
$\sigma(x,x')$. The coefficients of this series are determined by the chain of
auxiliary differential operators acting on the two-point parallel transport
tensor, which in their turn follow from the solution of special recursive
equations. The derivation of these operators and their recursive equations are
based on the covariant Fourier transform in curved spacetime. The series of
negative fractional powers of $\tau$ vanishes in the coincidence limit $x'=x$,
which makes the proposed method consistent with the heat kernel theory of
Seeley--Gilkey and generalizes it beyond the heat kernel diagonal in the form
of the asymptotic expansion in the domain $\sigma(x,x')\ll\tau^{1/M}$, $\tau\to
0$.$\nabla^a\sigma(x,x')\ll\tau^{1/2M}$, $\tau\to 0$.
