### Algebraic $K$-theory of the first Morava $K$-theory

For a prime $p\ge 5$, we compute the algebraic $K$-theory modulo $p$ and ${v}_{1}$ of the mod $p$ Adams summand, using topological cyclic homology. On the way, we evaluate its modulo $p$ and ${v}_{1}$ topological Hochschild homology. Using a localization sequence, we also compute the $K$-theory modulo $p$ and ${v}_{1}$ of the first Morava $K$-theory.