The issues will be quick hacked (fixed if you wish) running the following SQL update statement on the Data.StackExchange database by someone with full database access.
That sites table does get/should be populated now and then (on changes) from a different source so maybe it is wiser to fix that instead.
I have tested below script here
Use [Data.StackExchange]
-- writing
update Sites
set BadgeIconUrl = 'https://cdn.sstatic.net/Sites/writing/img/apple-touch-icon.png'
, ImageUrl = 'https://cdn.sstatic.net/Sites/writing/img/logo.png'
, IconUrl = 'https://cdn.sstatic.net/Sites/writing/img/icon-16.png'
, Longname = 'Writing'
where Id = 84
-- writing Meta
update Sites
set BadgeIconUrl = 'https://cdn.sstatic.net/Sites/writingmeta/img/apple-touch-icon.png'
, ImageUrl = 'https://cdn.sstatic.net/Sites/writingmeta/img/logo.png'
, IconUrl = 'https://cdn.sstatic.net/Sites/writingmeta/img/icon-16.png'
, Longname = 'Writing Meta'
where Id = 85
-- CosSci
update Sites
set BadgeIconUrl = 'https://cdn.sstatic.net/Sites/psychology/img/apple-touch-icon.png'
, ImageUrl = 'https://cdn.sstatic.net/Sites/psychology/img/logo.png'
, IconUrl = 'https://cdn.sstatic.net/Sites/psychology/img/icon-16.png'
, Longname = 'Psychology & Neuroscience'
where Id = 146
-- CogSci Meta
update Sites
set BadgeIconUrl = 'https://cdn.sstatic.net/Sites/psychologymeta/img/apple-touch-icon.png'
, ImageUrl = 'https://cdn.sstatic.net/Sites/psychologymeta/img/logo.png'
, IconUrl = 'https://cdn.sstatic.net/Sites/psychologymeta/img/icon-16.png'
, Longname = 'Psychology & Neuroscience Meta'
where Id = 147
For reference here is what the icons will look like:
Writing
BadgeIconUrl

ImageUrl

IconUrl

writing Meta
BadgeIconUrl

ImageUrl

IconUrl

CogSci
BadgeIconUrl

ImageUrl

IconUrl

CogSci Meta
BadgeIconUrl

ImageUrl

IconUrl
