Interesting, wouldn’t a strike of the caliber you describe render any payment delivery system down for quite some time ? especially S.S.payments being as they are all delivered electronically?
EMP is most effective with high altitutde air blasts, which is part of the Russia restraint in using tactical nuke air blasts in Ukraine IMO. Those seeking destruction would mostly likely target ground blasts for mass fallout and deaths. I believe there is a possibility of World Wide Web (WWW) and banking survival in the nation, which allong with the SSA COP site, would be the key survivability issues. Despite Hollywood performances, SSA checks don't' "come in the mail." As of March 2013, they had to go to electroniccally to direct deposit in banks.
With nuclear weapons,
Surface-burst EMP (SBEMP) is different from High-altitude EMP (HEMP), so altitude is everything.
According to GlobalSecurity.org, "Because of the physics involved, SBEMP fields extend only to ranges on the order of 10 to 20 kilometers from the point of detonation." "SBEMP from surface bursts at altitudes of about 200 meters or less can be even more powerful than HEMP--HEMP energy levels may be in the range of 50,000 volts per meter. Values for SBEMP may be in the region of one million volts per meter. Furthermore, these high levels of SBEMP may couple (weld together) buried or above-ground cables outside the immediate vicinity of the nuclear detonation. If this happens, these cables may remain intact and transmit tremendous surges of energy to connected systems down the line." But for a 10 kt bomb, let us assume (as they did here) that the SBEMP extended 5000 meters (3 miles). So if we multiply that by 100 or 200 (which may not be true at all), we still only get 300 or 600 miles of SBEMP impact in that one attack out of 3.797 million mi².
As most of our nuclear scientists have dismissed the USSR-level threat for past 30 years, we don't have good models on Russia threat (the few that do work to underestimate that threat); while they have been focusing on North Korea and terrorists, with KT nukes. But it is possible that USA SSA could actually survive and continue to pay out to those dependent on Social Security payments, who have WWW access and are survivors.
So the only question would be for USA survivors, would WWW access continue and if so, where. You have to remember what makes USA a uniquely desirable target for Russia is its (relatively) large land mass 3.797 million mi² almost comparable to PRC (Russia =6.602 million mi²). That generally allows for both mass casualties and control of destruction largely to a single country.
HHS SSA does have a COP operations site (which I won't specifically identify here). So the destruction and irradiation of DC and probable NE fallout contour area does not automatically shut the SSA system down.
HHS USA SS payments are almost all electronic directly to banks, as of March 01, 2013. I remember going to the local office with the my mother in Pennsylvania (long since passed now), and seeing armed guards in the lobby ready to deal with any violent protests. So as long as the COP location could get the SSA payments to the banks (that survive), people could continue to get Social Security payments.
Edit - it occurs to me that there could be readers that don't know the acronym WWW for World Wide Web, as thinking that way is considered a dated concept. But course, there are not only Internet backbone networks, but also the WWW interface which is the key usability function of the Internet for millions.