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The paper is concerned with $p$-adic $L$-functions (in partially $p$-adic zeta functions) and cyclotomic $p$-adic (multiple) zeta values.

Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov and others. S. Unver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple zeta values is given by Furusho.

Results on multiple zeta values, Galois groups and geometry of modular varieties has presented by Goncharov.

Interesting unipotent motivic fundamental group is defined and investigated by Deligne and Goncharov .

Tannakian interpretation of $p$-adic multiple zeta values is given by Furusho.

In our communication we will consider in frameworks of $p$-adic $L$-functions and $p$-adic (multiple) zeta values the application of approaches by Kubota-Leopoldt and by Iwasawa which are based on Kubota-Leopoldt $p$-adic $L$-functions and arithmetic $p$-adic $L$-functions by Iwasava.

Voronoi-type congruences for corresponding Bernoulli numbers will be presented.

Numerical examples are included.