  We use a tensor network renormalization group method to study random $S=2$
antiferromagnetic Heisenberg chains with alternating bond strength
distributions. In the absence of randomness, bond alternation induces two
quantum critical points between the $S=2$ Haldane phase, a partially dimerized
phase and a fully dimerized phase, depending on the strength of dimerization.
These three phases, called ($\sigma$,$4-\sigma$)=(2,2), (3,1) and (4,0) phases,
are valence-bond solid (VBS) states characterized by $\sigma$ valence bonds
forming across even links and $4-\sigma$ valence bonds on odd links. Here we
study the effects of bond randomness on the ground states of the dimerized spin
chain, calculating disorder-averaged twist order parameters and spin
correlations. We classify the types of random VBS phases depending on the
strength of bond randomness $R$ and dimerization $D$ using the twist order
parameter, which has a negative/positive sign for a VBS phase with odd/even
$\sigma$. Our results demonstrate the existence of a multicritical point in the
intermediate disorder regime with finite dimerization, where (2,2), (3,1) and
(4,0) phases meet. This multicritical point is at the junction of three phase
boundaries in the $R$-$D$ plane: the (2,2)-(3,1) and (3,1)-(4,0) boundaries
that extend to zero randomness, and the (2,2)-(4,0) phase boundary that
connects another multicritical point in the undimerized limit. The undimerized
multicritical point separates a gapless Haldane phase and an
infinite-randomness critical line with the diverging dynamic critical exponent
in the large $R$ limit at $D=0$. Furthermore, we identify the (3,1)-(4,0) phase
boundary as an infinite-randomness critical line even at small $R$, and find
the signature of infinite randomness at the (2,2)-(3,1) phase boundary only in
the vicinity of the multicritical point.
