Correct spelling, explanation: secondary is a modern version of 14th-century secondarie, coming from Old French secondaire. Going even earlier we meet the Latin word secundus, meaning second (ordinal number). Knowing the root second we can create the adjective secondary and we don’t have any reason to change o to u in the root part. Therefore secundary is incorrect and the correct form is secondary.

Definition of secondary:
adjective, coming after, as a second, not so important; also: secondary school (after primary school)
While the main character of the story was compelling, the secondary characters also played an important role in advancing the plot.
The primary goal of the project was to improve customer satisfaction, but a secondary objective was to reduce costs.

Collocations with secondary:
secondary education, education that follows primary education
In many countries, secondary education is obligatory for all students under the age of 18.
secondary colours, colours outside of the primary colours, created by mixing primary colours
Mixing yellow and blue creates the secondary color green, which is commonly associated with nature and growth.

Incorrect spelling, explanation: this form is incorrect, due to the fact, that the root noun second is spelled with o and there is no need to change it to u, or any other letter when creating an adjective secondary. Secundary is therefore incorrect; it doesn’t make sense in pronunciation as well – it does not indicate such spelling. Secondary is the only correct form here.

Incorrect spelling

Incorrect spelling

Incorrect spelling