Spikes like this in reported cases are not unusual, see also this graph for China and the explanation for it. There can be various reporting "bottlenecks" that suddenly report backlogged tests.
In France, what seems to have happened on April 3-4 (as Reuters reported):
The number of people who have died from the COVID-19 illness in France jumped by 61% to 6,507 over two days after data from nursing homes were included, and confirmed cases jumped by 44% to 82,165, the health ministry reported on Friday, making France the fifth country to report more cases than China.
Salomon also reported a total of 17,827 confirmed or suspected cases in nursing homes, compared to 14,638 on Thursday, when the nursing home data were first reported.
By including the nursing home data, France now has a total of 82,165 confirmed or suspected cases. It joined the United States, Spain, Italy and Germany as nations that have now surpassed cases reported by China, where the pandemic originated, according to a Reuters tally. [...]
Following criticism about French COVID-19 statistics not reflecting the dozens of people dying in senior citizens’ residences or their own homes, the health ministry on Thursday provided the first data on deaths in nursing homes.
A far as JHU vs official santepubliquefrance.fr (SP.FR henceforth) is concerned, they indeed appear to diverge on "total cases", but the explanation for that is that the left upper "cyan box" in the French dashboard ("confirmed cases") is still showing just the hospital-confirmed cases. You need to add the bottom right, dark blue box "CAS en ESMS" for the cases in nursing homes, which also has a sub-field for deaths therein. (Thanks to @Relaxed for pointing out this; a previous version of this answer incorrectly said that you needed to add the "closed" hospital cases [return home] (green upper-right box), but while those numbers were closer to the ESMS ones yesterday, today there's 10K difference today, which makes my prior explanation clearly wrong in that respect.)
This can be double-checked in a way by looking at a 3rd source, worldometers.info, which cites SP.FR as their data source, but presents the info much more like JHU, i.e. worldometers adds the ESMS cases to the hospital ones. Three large screencaps follow:
So, SP.FR "total cases" = hospital-confirmed (78,167) + ESMS (30,902) = 109,069 == worldometers total. JHU total: 110,070 (pretty close). Deaths: SP.FR "red box" 10,328 == worldometers deaths. JHU deaths: 10,343, pretty close. ("Recoverd" case, i.e. returns from hospital is also largely in agreement 19.5K JHU vs 19.3K SP.FR/worldometers)
The same worldometers page also makes it clear that the (Apr 3) Reuters-reported spike was real/substantial (that graph is from yesterday):
So in summary: yes there was a spike in reported cases (even by the French) due a change in their methodology (including nursing homes) on April 3, and additionally the SP.FR page still separates total cases in hospital-confirmed and ESMS ones...
(I'm still not sure exactly how SP.FR adds the deaths since the ESMS death sub-field is 4K, but their "red box" has 7K hospital deaths... while the red-box total is only 10.3K.)