"I read the news on my computer."
Just to be 100% correct, I think that the sentence "I read the news on my computer" should be actually translated as "我用我的电脑看新闻。"
This sentence can be read in two ways in english: 1. past tense 2. or as if it is stating how you read the news regularly.
Hover-over was of absolutely no value.
I'm definitely not an expert in Chinese, but I think putting the completed action marker 了 immediately after the 用 would make the sentence past tense.
I am not a native speaker, I believe 我用电脑看了新闻 would be past tense (I have read). You would need 了 after 看. If you put 了 after 用， it would be 'I used a computer'. The action being completed is 'reading'.
Is it implied here that the computer is mine or should it be stated explicitly?