Hi Lara!
This is my first ever review on these forums, so please bear with me.
This was actually very well written.

I can see what you're trying to do with the varying perspectives, but I think it gets just a bit confusing. I don't know what the rest of the story is going to be like, obviously, but I would suggest maybe doing one character's point of view for a chapter, and then switching it in the next one if you want to change perspectives. Or perhaps you can use something to show that the perspective has now switched (a different font perhaps? I'm not really sure).
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He had his bottom lip tucked in, he looked like a child, concentrating hard.
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This just sounds a bit awkward to me. Perhaps try something more along the lines of:
He captured his bottom lip between his teeth. It made him look like a child who was concentrating hard on his task. Not the best example, I know.
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“I thought we could go away for a few days. Maybe visit your sister. You said you would like to see her before the baby is born” he said as he tugged at the sleeve of his jumper.
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This reads just a tad awkwardly. I would suggest something along the lines of:
"I thought we could go away for a few days," he said as he tugged at the sleeve of his jumper. "Maybe visit your sister. You said you would like to see her before the baby is born."
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“Alright” he muttered without looking up. He was watching a storm in his coffee cup.
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I think that should be
"Alright," he muttered... And...I'm not quite sure what you mean by the storm in his coffee cup. Do you mean that he's gazing intently into his coffee? If so, maybe you should say something a bit more along those lines, because it's a tad confusing.
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“Why” she began to ask. Smoke and tar filled the gap between us. “Why, do we always come here?”
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Again, a comma after the first 'why' would be better, and you can remove the comma from the second 'why'.
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She looked around. A bored expression on her face.
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You can make that all one sentence:
She looked around, a bored expression on her face.
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“Maybe we could go somewhere else, why don’t we leave here.
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I think I would split that up into two sentences:
"Maybe we could to somewhere else. Why don't we leave here?"
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Deep lines had formed from the corner of his eyes.
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Maybe change 'from' to 'in'
I understand that you left the paragraph breaks solely for showing when the perspective shifts, but as I said above, maybe you can find a different way of showing those changes in views so that you can add in the paragraph breaks. It's just all-around easier to read that way.
Anyway...all of this is basically just really nit-picky things. I think you did a great job of getting into the character's heads (which is always important) and in the dialogue between the two. You already have me interested about who this woman is who died - his mother? A sister? An ex-lover? And it will be interesting to see where their conversation takes them now.