Ure is Scottish, but he speaks standard British English. In Britain, every area has a different accent, but educated people from anywhere can have an RP accent (the standard British pronunciation). This accent is not particularly associated to an area (though the closest local accent is Oxford), so you can find RP speakers in any part of England, Wales, Scotland or Ireland. If you pay attention to Ure's pronunciation, you will possibly notice that the only Scottish thing he's got to his accent is the way he pronounces the diphthong / eɪ /.

BREATHE= To let air in and out of your body. Compare the pronunciation of the verb to BREATHE /bri:ð/ and the noun BREATH /breθ/. Notice that not only does* the pronunciation of the vowel change, but also the final consonant.    [*a sentence beginning with "not only" usually has an inversion, like questions]

WHAT LIFE HAS DEALT TO ME= When you play cards, to deal the cards means to give cards to each player, so what life has dealt to him refers to his luck, his fate, his present situation, a situation that he has passively received.

DENY= If you deny something you say it’s not true or you simply don’t accept it.

CHANCE= Opportunity. The standard British pronunciation is /tʃɑ:ns/ but in American and many parts of Britain it is pronounced /tʃæns/ (like in this song).
WITH EVERY SADNESS I DENY, I FEEL A CHANCE INSIDE ME DIE= To live a full life, you must accept the good and the bad things. Only if you know sadness, can you* appreciate joy.   [*a sentence beginning with "only" usually has an inversion, like questions]

A TASTE= The act of tasting or a small quantity to taste, a sample. To taste is to eat or drink something to appreciate it. Here he uses this word metaphorically, so "a taste of something new" means "a new experience".

HOLD= To embrace, to hug, to put your arms around a person or thing. Used metaphorically here.

TO PULL ME THROUGH= To take me away from this bad situation.

THIS DARKENED LIFE OF MINE= We can’t use a possessive adjective in front of a determiner (articles, demonstratives, etc.), so we use this other construction with "of + possessive pronoun":
- He’s my friend / take my book
- He’s a friend / take this book
- He’s my a friend / take my this book (wrong) He’s a friend of mine / Take this book of mine.
So if we want to mix "across this life" with "across my life" we have to say "across this life of mine".

BREATHE SOME SOUL IN ME= Put some soul inside of me. He uses the verb "breathe" instead of "introduce" because it makes it sound much more gentle and loving. This also brings about biblical references: in the book of Genesis, when God created man he made a body with mud and then breathed life into it, or more exactly, "He breathed some soul into him".

GIFT= Present.

‘FORE= Before.

BREATHE LIFE TO LAY ‘FORE ME= Put some life in front of me, that is, he’s very sad and he wants to have again some future, some hope for life (when you are depressed you can see no future ahead).

FOR EVERY MAN WHO BUILT A HOME= More biblical references. In the New Testament, Jesus tells the parable of "the wise and foolish builders". A wise man built his house on stone, and a foolish man built his house on sad. Then came the waters and destroyed the house built on sand. (Mat 7:24-27). In this song, the flow of water of the parable becomes "an open flow of lies and failiures", so, as in the parable, only the man who built his life on solid grounds can successfully fight against all that.

GIVE ME A TOUCH OF SOMETHING SURE= He’s asking for faith, for certainties.

EVERMORE= Forever.

THE HIGHEST HIGHS CAN TURN AROUND= This refers to a huge and unexpected change of situation. HIGH is usually an adjective but he is using it as a noun (that's why it's in the plural form), so HIGHS means "high places".

STONY= Made of stone.

TO SOW LOVE’S SEEDS ON STONY GROUND= Here, he may be referring to the past (how his life was destroyed) or to the future (hope for new better things). If he refers to the past, this sentence has, again, biblical references. Another parable from Jesus talks about the sower (Mar 4:3-8) who sowed seeds on different grounds. The seed that fell on rocky grounds died without bearing fruit. But he might be using this sentence to talk about the power of God, who can make seeds grow even on stony ground. I, personally, don’t know if this paragraph is talking about the past or the future, so both interpretations could make sense. You choose.

BREATHE YOUR INNOCENCE TO ME= He wants to be honest, innocent, etc, so he’s asking God for those things. If Gods breathes innocence to him, he will be filled with God’s innocence and he'll be innocent.
Notice that in English double consonants are always pronounced like a single one, so INNOCENCE sounds /ɪnəsəns/. If we wrote this with only one N, then the previous vowel would be a diphthong: "inocence" would sound /aɪnəsəns/ (more info), so doubling consonants affects vocalic sounds, not the consonants themselves.