The SPA sample of Senegal 2012-2013 was a stratified sample. In each region, the Health Facilities (HF) were classified by crossing the management authority (Government/private) with the health facility type (Hospitals, Health Centers and Health Posts). The approximate sampling fraction at national level was given in the final report as you cited. But at sampling stratum level, the fraction varies a little depending on availability of HF.

As for the client weight (sick child or family planning) in the 2012-2013 survey, because it was not able to link the client with the provider, the provider weight is not taken into count. The client weight uses the HF weight as base weight.
The SPA survey still evolves, the sampling weight calculation has been changed since the 2012-2013 survey. A general provider weight was calculated (not taken the provider type into account) as base weight for clients which we believe improves a little the client weight. Since the field procedure will not guarantee that for every interviewed client, their provider is interviewed, so the client weight is not an "exact" weight but an approximate weight. Hope this will continue to improve if there is an exact "interviewed provider"-"interviewed client" relation for all interviewed clients.

Hope this is helpful.