In general Lord Shiva loves Anna(rice) for Naivedya and also anna daanas(charity).
This chapter of Linga Purana states:
As for Naivedya, ‘Shuddhhaanna’ would satisfy Maha Deva; in fact,
‘Anna Daana’ is one significant charity to the needy that Shiva is
always desirous of on any day, especially on Pouranamasi.
If this anna abhishekam is done every poornima [full moon] or on selected months poornima only? If so which months?
The above passage shows that Lord Shiva loves anna charity specially on Purnimas.The following passage from the same Purana shows that it is of special significance in the Karthika month:
Kartika Month: The entire month is of signifinance and daily
Shivaarchana-Abhisheka with Sacred water, Milk, Coconut water, Curd,
Ghee, Chandana, and Bhasma with Maha Nyasa Purvaka, Maha Rudra Purvaka
Snaanaas to Shiva Linga, Day long fasting and Alpaahara in the nights,
complete abstinence, Bhu Shayana and various sacrifices should be in
place; all kinds of Daanaas on anyday or as many days are encouraged
in the Month and similarly Pandita Seva of any kind would be fruitful
in the Month. Emphasis is stressed on Anna Daanaas during the month.
Shiva Pujas ,Rudraabhisekhams are of special significance on Purnima Tithis in general & not just on the Karthika Purnima.
The rewards of performing Shiva Puja,Abhisekhams ,Vratas etc on Purnimas of various months,given in this Linga Purana chapter, are as follows:
Shiva Vratas were also prescribed by Nandi Deva to Skanda as follows:
Pushya Purnami: Shiva Pujaabhisheka with Ghrita, Ksheera, Dadhi etc followed by Brahmana bhojana and Godaana , Ratri bhojana;
Phala: Brahmaloka Prapti.
Chaitra Purnima: Shiva Pujaabhisheka, Raatri bhojana of rice and ghee, Brahmana Bhojana and daana of two white cows
Vaishakha Purnami: Shiva Pujaabhisheka with Panchagavyas, Ratri bhojana, Brahmana Bhojana and Go-Daana; Phala: Ashwamedha Yagna
Jeyshtha Purnima: Shiva Pujaabhisheka, Ratri Bhojana, Brahmana Seva and Dhumravarna Godaana; Phala: Vaayu loka prapti
Ashadha Purnima: Shiva Pujabhisheka, Ratri bhojana, Brahmana Seva, Gaura varna Godaana; Phala: Varuna Loka
Shravana Purnima: Shiva Pujaabhisheka, Ratri Bhojana, Vrushabha daana; Phala:Vayu loka
Bhadrapada Purnima: Havana, Abhishka and Puja; Veda Parayana, Brahmana puja, Vrishabha dwaya daana Phala: Yaksha loka Rajastwa.
Ashviyuja Purnima: Shiva Pujaabhisheka, day long fast, Brahmana
Bhojana and Nila Varna Vishabha daana. Phala: Ishana loka prapti.
Kartika Month: The entire month is of signifinance and daily Shivaarchana-Abhisheka with Sacred water, Milk, Coconut water, Curd,
Ghee, Chandana, and Bhasma with Maha Nyasa Purvaka, Maha Rudra Purvaka
Snaanaas to Shiva Linga, Day long fasting and Alpaahara in the nights,
complete abstinence, Bhu Shayana and various sacrifices should be in
place; all kinds of Daanaas on anyday or as many days are encouraged
in the Month and similarly Pandita Seva of any kind would be fruitful
in the Month. Emphasis is stressed on Anna Daanaas during the month.
Phala: Surya Loka Prapti.
Margasirsha Month is considered an an extension of the previous month when again austerities would spill over from the previous month;
Upavaasaas, Shiva worship by way of Abhishekas, Brahmana Sevas, Anna
and other kinds of daanaas especially to Daridras, would be of
significance and the Phala would be of Chandra loka prapti.