  We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938),
Ernvall-Metsankyla (1991) on the sums of powers of integers weighted by powers
of the Fermat quotients to the next Fermat quotient power, namely to the third
power of the Fermat quotient. Using this result and the Gessel identity (2005)
combined with our past work (2001), we are able to relate residues of some
truncated convolutions of Bernoulli numbers with some Ernvall-Metsankyla
residues to some full convolutions of the same kind. We also establish some
congruences concerning other related weighed sums of powers of integers when
these sums are weighted by some analogs of the Teichmuller characters.
