I did check the system log - apparently there is something wrong with sandbox, but I have no idea what is sandbox and what is it doing...
Jul 11 11:36:39 iMac.local storeassetd[355]: addOperation <UpdateOperation: 0x7fc7d8574a20>{name = '(null)'}
Jul 11 11:36:39 iMac.local storedownloadd[354]: DownloadServiceDelegate: Accepting new connection <NSXPCConnection: 0x7fc78b423f50> connection from pid 355 with interface <DownloadServiceInterface: 0x7fc78b444b50> (PID 355)
Jul 11 11:36:39 iMac.local storeaccountd[329]: AccountServiceDelegate: Accepting new connection <NSXPCConnection: 0x7fa82630ade0> connection from pid 461 with interface <AccountServiceInterface: 0x7fa823e772f0> (PID 461)
Jul 11 11:36:42 iMac.local genatsdb[698]: sandbox cache error 8: attempt to write a readonly database
Jul 11 11:36:43 iMac.local genatsdb[698]: SQLite: unixTruncateDatabase() failed with 10 for replacing '/private/var/folders/zz/zyxvpxvq6csfxvn_n00000c4000031/C/com.apple.sandbox/san dbox-cache.db' with flags 0
Jul 11 11:36:43 iMac.local genatsdb[698]: sandbox cache could not be reset
Jul 11 11:36:43 iMac.local sandboxd[350] ([232]): softwareupdated(232) deny file-write-create /private/var/folders/zz/zyxvpxvq6csfxvn_n00000s0000068/C/com.apple.SoftwareUpda te.SUUpdateServiceDaemon.state.sb-b011cf5e-d7FHu7
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Harrington', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Tw%20Cen%20MT%20Bold.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Monotype%20Corsiva', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/PMingLiU-ExtB.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/SimSun-ExtB.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Arial%20Italic.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Cambria%20Italic.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Onyx', skipping file
****************************************** (many more of the same type message - missing fonts mostly that I deleted)**********************************
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Copperplate%20Gothic%20Light', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Meiryo%20Bold.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Corbel%20Bold%20Italic.ttf', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Perpetua%20Titling%20MT', skipping file
Jul 11 11:36:43 iMac.local fontd[696]: Cannot save bookmark for file 'file:///Library/Fonts/Microsoft/Consolas%20Bold.ttf', skipping file
Jul 11 11:36:44 iMac.local fontd[696]: ATS is unable to switch to Read/Write database - ATS will run off of a read-only database. Only System Fonts will be available.
Jul 11 11:36:46 iMac com.apple.xpc.launchd[1] (org.postfix.master[699]): Service exited with abnormal code: 1
I found similar topic here (planning on trying the solution):
cannot install updates from app store