  We study a modified record process where the $k$'th record in a series of
independent and identically distributed random variables is defined recursively
through the condition $Y_k > Y_{k-1} - \delta_{k-1}$ with a deterministic
sequence $\delta_k > 0$ called the handicap. For constant $\delta_k \equiv
\delta$ and exponentially distributed random variables it has been shown in
previous work that the process displays a phase transition as a function of
$\delta$ between a normal phase where the mean record value increases
indefinitely and a stationary phase where the mean record value remains bounded
and a finite fraction of all entries are records (Park \textit{et al} 2015 {\it
Phys. Rev.} E \textbf{91} 042707). Here we explore the behavior for general
probability distributions and decreasing and increasing sequences $\delta_k$,
focusing in particular on the case when $\delta_k$ matches the typical spacing
between subsequent records in the underlying simple record process without
handicap. We find that a continuous phase transition occurs only in the
exponential case, but a novel kind of first order transition emerges when
$\delta_k$ is increasing. The problem is partly motivated by the dynamics of
evolutionary adaptation in biological fitness landscapes, where $\delta_k$
corresponds to the change of the deterministic fitness component after $k$
mutational steps. The results for the record process are used to compute the
mean number of steps that a population performs in such a landscape before
being trapped at a local fitness maximum.
