  Engine-driven explosions with continuous energy input from the central system
have been suggested for supernovae (SNe) associated with a Gamma-Ray Burst
(GRB), super-luminous SNe (SLSNe), and at least a fraction of broad-lined SNe
Ic (SNe Ic-BL) even without an associated GRB. In the present work, we
investigate observational consequences in this scenario, focusing on the case
where the energy injection is sufficiently brief, which has been suggested for
GRB-SNe. We construct a simplified, spherical ejecta model sequence taking into
account the major effects of the central engine; composition mixing, density
structure, and the outermost ejecta velocity. Unlike most of the previous works
for GRB-SNe, we solve the formation of the photosphere self-consistently, with
which we can predict the photometric and spectroscopic observables. We find
that these ejecta properties strongly affect their observational appearance in
the initial phase (~ a week since the explosion), highlighted by blended lines
suffering from higher-velocity absorptions for the flatter density distribution
and/or higher outermost ejeca velocity. This behaviour also affects the
multi-band light curves in a non-monotonic way. Prompt follow-up observations
starting immediately after the explosion thus provides key diagnostics to
unveil the nature of the central engine behind GRB-SNe and SNe Ic-BL. For SN
2017iuk associated with GRB 171205A these diagnosing observational data are
available, and we show that the expected structure from the engine-driven
explosion, i.e., a flat power-law density structure extending up to >~ 100,000
km/s, can explain the observed spectral evolution reasonably well.
