Let
K
\mathbb {K}
be a function field of characteristic
p
>
0
p>0
. We have recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over
K
(
z
)
\mathbb {K}(z)
. This paper is dedicated to proving the following refinement of this theorem. Let
f
1
(
z
)
,
…
,
f
n
(
z
)
f_{1}(z),\ldots , f_{n}(z)
be
d
d
-Mahler functions such that
K
¯
(
z
)
(
f
1
(
z
)
,
…
,
f
n
(
z
)
)
\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )
is a regular extension over
K
¯
(
z
)
\overline {\mathbb {K}}(z)
. Then, every homogeneous algebraic relation over
K
¯
\overline {\mathbb {K}}
between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over
K
¯
(
z
)
\overline {\mathbb {K}}(z)
between these functions themselves. If
K
\mathbb {K}
is replaced by a number field, this result is due to B. Adamczewski and C. Faverjon as a consequence of a theorem of P. Philippon. The main difference is that in characteristic zero, every
d
d
-Mahler extension is regular, whereas in characteristic
p
p
, non-regular
d
d
-Mahler extensions do exist. Furthermore, we prove that the regularity of the field extension
K
¯
(
z
)
(
f
1
(
z
)
,
…
,
f
n
(
z
)
)
\overline {\mathbb {K}}(z)\left (f_{1}(z),\ldots , f_{n}(z)\right )
is also necessary for our refinement to hold. Besides, we show that when
p
∤
d
p\nmid d
,
d
d
-Mahler extensions over
K
¯
(
z
)
\overline {\mathbb {K}}(z)
are always regular. Finally, we describe some consequences of our main result concerning the transcendence of values of
d
d
-Mahler functions at algebraic points.