  Certain carbon-enhanced metal-poor stars likely obtained their composition
via pollution from some of the earliest generations of asymptotic giant branch
stars and as such provide important clues to early Universe nucleosynthesis.
Recently, Kinman et al. discovered that the highly carbon- and barium-enriched
metal-poor star SDSS J1707+58 is in fact an RR Lyrae pulsator. This gives us an
object in a definite evolutionary state where the effects of dilution of
material during the Main Sequence are minimised owing to the object having
passed through first dredge-up. We perform detailed stellar modelling of
putative progenitor systems in which we accreted material from asymptotic giant
branch stars in the mass range 1-2 solar masses. We investigate how the surface
abundances are affected by the inclusion of mechanisms like thermohaline mixing
and gravitational settling. While we are able to find a reasonable fit to the
carbon and sodium abundances of SDSS J1707+58, suggesting accretion of around
0.1 solar masses from a 2 solar mass companion, the strontium and barium
abundances remain problematic and this object may have experienced something
other than a standard $s$ process. We have more success in fitting the
abundances of the mildly carbon-enriched, metal-poor RR Lyrae pulsator TY Gru
(CS 22881-071), which we suggest received 0.1 solar masses of material from a
companion of around 1 solar mass.
