It’s grammatically incorrect and sounds awkward to native English speakers. The word “news” is one of those words that sounds plural (meaning there’s more than one) but is actually singular (there’s only one). It’s called a mass noun and is usually used with a singular verb.
In English, most nouns add “-s” or “-es” to the end of the word to make it plural; cat becomes cats, house becomes houses. Sometimes, the entire word changes in the plural: mouse becomes mice, goose becomes geese. Some words can be plural or singular, depending on the context— moose is one example that can be one animal or a whole herd.
So, back to your question. “News” is like the word “scissors” or “trousers”. It’s generally understood that although it sounds plural, it’s actually singular. These kinds of nouns are called ‘plurale tantum‘.
(Plurale tantum is a noun that appears only in the plural and doesn’t ordinarily have a singular form (for example, jeans, pyjamas, tweezers, shears, and scissors). Also known as a lexical plural. Plural: pluralia tantum. Jeans, scissors, trousers and glasses are great examples of plural tantum nouns in the English language. There is, of course, the opposite as well. Singular tantum, a noun that appears only in the singular form, such as dirt, is known as singulare tantum.)
In order to use the article/adjective “a”, you’d need to change the way you phrase it.
“I’ve got a bit of good news.”
“I have a lot of good news.”
Otherwise, the correct way to say it is, “I have good news.”