  The recently obtained hairy Kerr black holes, due to additional sources or
surrounding fluid, like dark matter, conserving energy-momentum tensor, are
with a deviation $\alpha$, and primary hair $l_0$ apart from rotation parameter
$a$ and mass $M$. The Event Horizon Telescope (EHT) has released the first
image of supermassive black hole M87*, revealing that the observed image is
consistent with the shadow of a Kerr black hole and has opened up a new window
to probe the strong gravity regime. In the wake of the EHT observations, the
recent interest in black hole shadow suggests comparing the black holes in
general relativity (GR) and modified theories of gravity to assess these
models' differences. Motivated by this, we consider the rotating hairy Kerr
black holes, which encompasses, in particular cases, Kerr ($\alpha=0$) black
holes. Interestingly, for a set of parameters ($M$, $a$, and $\alpha$), there
exists an extremal value of the hair parameter $l_E$, which corresponds to an
extremal black hole with degenerate horizons. In contrast, for $l_0>l_E$, it
describes a nonextremal black hole with Cauchy and event horizons and a naked
singularity for $l_0<l_E$. We investigate ergosphere and shadow of hairy Kerr
black holes to infer that their size and shape are affected due to the $l_0$
and have a richer chaotic structure. In particular, the hairy Kerr black holes
possess smaller size but more distorted shadows when compared with Kerr black
holes. We also estimate the parameters $l_0$ and $a$ associated with hairy Kerr
black holes using the shadow observables. We show that the inferred circularity
deviation $\Delta C \leq 0.1$ for the M87* black hole puts a constraint on
hairy Kerr black hole parameter $a$, whereas angular shadow diameter $\theta_d
= 42 \pm 3 \mu as$, within $1 \sigma$ region, for a given choice of $\alpha$,
also places bounds on these parameters.
