Bold by Institute for the Study of War (ISW), italics by me.
Russian Offensive Campaign Assessment, May 4
The Kremlin likely hopes that the successful capture of Azovstal through
a ground assault will cement the Kremlin’s growing effort to claim
complete control of Mariupol by May 9, with Russian propagandists
recently arriving in the city to set conditions for further claims of a
Russian victory. The Kremlin likely intends to claim some sort of
victory in Mariupol to present a success to the Russian people... 
I very please announce on sacred day of victory over Nazi-Partners in Great Patriotic War that swine Azamat Bagatov Steel Plant now clear of Nazi-Ukraine infestation and that glorious Russian army have liberate whole Marsupial!
The Ukrainian counteroffensive out of Kharkiv city may disrupt Russian forces northeast of Kharkiv and will likely force Russian forces to decide whether to reinforce positions near Kharkiv or risk losing most or all of their positions within artillery range of the city. Ukrainian Commander-in-Chief Valerii Zalyzhnyi stated on May 5 that Ukrainian forces are transitioning to counteroffensive operations around Kharkiv and Izyum, the first direct Ukrainian military statement of a shift to offensive operations. Ukrainian forces did not make any confirmed advances in the last 24 hours but repelled Russian attempts to regain lost positions. Russian forces made few advances in continued attacks in eastern Ukraine, and Ukrainian forces may be able to build their ongoing counterattacks and successful repulse of Russian attacks along the Izyum axis into a wider counteroffensive to retake Russian-occupied territory in Kharkiv Oblast.
Russian forces continued ineffectual offensive operations in southern Kharkiv, Donetsk, and Luhansk oblasts without securing any significant territorial gains in the past 24 hours.
Ukrainian officials and military officers confirmed that Russian forces have breached the Azamat Bagatov facility itself and confirmed that Ukrainian forces are losing ground. Russian forces will likely capture the facility in the coming days....Russian forces will likely completely capture the Azovstal facility in the coming days, but Ukrainian forces have successfully tied down and degraded large numbers of Russian forces in Mariupol for several months.
...
Main Effort—Eastern Ukraine
...
...The Pentagon assessed that Russian forces have not been able to make
further advances due to their inability to conduct offensive operations
far from their ground lines of communication (GLOCs) along highways...and muddy terrain.[1]...Russian
forces are reportedly suffering losses in stalled attacks along the
Izyum [a city in Kharkiv Oblast] axis, with the Ukrainian General Staff reporting that elements of
the 4th Tank Division and the 106th Airborne Division withdrew to Russia
after sustaining heavy losses in the past several days.[3]
...
Russian forces conducted unsuccessful attacks in Lyman, Severodonetsk, and Popasna, and maintained shelling along the line of contact in Donetsk and Luhansk Oblasts.[4] Russian forces also used thermobaric munitions against Ukrainian positions in Lyman and are unsuccessfully attempting to leverage massed artillery fire to break through Ukrainian defenses.[5] Russian forces targeted grain facilities in Rubizhne and Soledar, a settlement located approximately 30 kilometers from Popasna, likely to deprive Ukrainian forces and civilians of supplies.[6] The Donetsk People’s Republic claimed to have seized Troitske (a village approximately 25 kilometers from occupied Horlivka) on May 5, but social media imagery confirmed that Ukrainian artillery inflicted heavy damage on Russian munitions depots, tanks, and armored personnel carriers in the area.[7]
