One spoon is "łyżka". "potrzebować" takes Genitive, and the Genitive singular form is "łyżki".
Yes, just like plural Nominative and Accusative.
I need new spoons was rejected,lyzka is singular lyzki is plural, correct? Both words were in English but spoons plural was rejected.
Cases. In most cases, Genitive singular (what we have here) is identical to Nominative plural.
As you have the verb "potrzebować", which takes Genitive, you know that there is a singular spoon, because otherwise it would be "Potrzebuję nowych łyżek".